Osmania Universitq uorur M
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INTRODUCTION
QUANTUM MECHANICS
With Applications to Chemistry
LINUS PAUlteG, PhJ>>, Sc.D
Professor of Chemistry, California institute of TecfonMotflT
A#D
E. BRIGHT WILSON,, Jr., Ph.D.
Associate Professor of Chemistry, Harvard University
INTERNATIONAL STUDENT EDITION
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INTRODUCTION TO QUANTUM MECHANICS
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PREFACE
In writing this book we have attempted to produce a textbook
of practical quantum mechanics for the chemist, the experi
mental physicist, and the beginning student of theoretical
physics. The book is not intended to provide a critical discus
sion of quantum mechanics, nor even to presenita thorough
survey of the subject. We hope that it does glvek lucid and
easily understandable introduction to a limAdjporMon >»
quantummechanical theorv; n*mv>K\ *ha( i Jtwi ■ ■ U&*#
suggested by the rhiu« ( .*..■/. / ., ; > n !■,,., ffefrifetii^K of the
alseussion oithe Schrodinger wave equal twi .• ft 15ie* problems
which can be treated by means of it. The effort has been made
to provide for the reader a means of equipping himself with a
practical grasp of this subject, so that he can apply quantum
mechanics to most of the chemical and physical problems which
may confront him.
The book is particularly designed for study by men without
extensive previous experience with advanced mathematics, such
as chemists interested in the subject because of its chemical
applications. We have assumed on the part of the reader, in
addition to elementary mathematics through the calculus, only
some knowledge of complex quantities, ordinary differential
equations, and the technique of partial differentiation. It
may be desirable that a book written for the reader not adept
at mathematics be richer in equations than one intended for
the mathematician; for the mathematician can follow a sketchy
derivation with ease, whereas if the less adept reader is to be
led safely through the usually straightforward but sometimes
rather complicated derivations of quantum mechanics a firm
guiding hand must be kept on him. Quantum mechanics is
essentially mathematical in character, and an understanding
of the subject without a thorough knowledge of the mathematical
methods involved and the results of their application cannot be
obtained. The student not thoroughly trained in the theory
of partial differential equations and orthogonal functions must
iv PREFACE
learn something of these subjects as he studies quantum mechan
ics. In order that he may do so, and that he may follow the
discussions given without danger of being deflected from the
course of the argument by inability to carry through some minor
step, we have avoided the temptation to condense the various
discussions into shorter and perhaps more elegant forms.
After introductory chapters on classical mechanics and the
old quantum theory, we have introduced the Schroding^r wave
equation and its physical interpretation on a postulatory basis,
and have then given in great detail the solution of the wave
equation for important systems (harmonic oscillator, hydrogen
atom) and the discussion of the wave functions and their proper
ties, omitting none of the mathematical steps except the most
\}?JWJ*tesr?7 A, similarly detailed treatment has been given
in the discussion o\ pert in 1 option 'ihreurj, the variation method,
the structure of simple molecules, and, in general, iij. •,
important section of the book.
In order to limit the size of the book, we have omitted from
discussion such advanced topics as transformation theory and
general quantum mechanics (aside from brief mention in the
last chapter), the Dirac theory of the electron, quantization
of the electromagnetic field, etc. We have also omitted several
subjects which are ordinarily considered as part of elementary
quantum mechanics, but which are of minor importance to the
chemist, such as the Zeeman effect and magnetic interactions in
general, the dispersion of light and allied phenomena, and
most of the theory of aperiodic processes.
The authors are severally indebted to Professor A. Sommerfeld
and Professors E. U. Condon and H. P. Robertson for their
own introduction to quantum mechanics. The constant advice
of Professor R. C. Tolman is gratefully acknowledged, as well
as the aid of Professor P. M. Morse, Dr. L. E. Sutton, Dr.
G. W. Wheland, Dr. L. 0. Brockway, Dr. J. Sherman, Dr. S.
Weinbaum, Mrs. Emily Buckingham Wilson, and Mrs. Ava
Helen Paulh^g.
Linus Pauling.
E. Bright Wilson, Jr.
Pasadena, ^alif.,
Cambridge Mass.,
July, 1935.
CONTENTS
Page
Preface , iii
Chapter I
SURVEY OF CLASSICAL MECHANICS
Section
1. Newton's Equations of Motion in the Lagrangian Form 2
la. The Threedimensional Isotropic Harmonic Oscillator. . 4
16. Generalized Coordinates \. A . . . . 6
lc. The Invariancc of the Equations of Motion in the Lagraib
gian Form 7
Id. An Example: The Isotropic Harmonic Oscillator in Polar
Coordinates 9
lc. The Conservation of Angular Momentum . . . 11
2. The Equations of Motion in the Hamiltonian Form 14
2a. Generalized Momenta 14
26. The Hamiltonian Function and Equations 16
2c. The Hamiltonian Function and the Energy 16
2d. A General Example 17
3. The Emission and Absorption of Radiation . . 21
4. Summary of Chapter 1 23
Chapter II
THE OLD QUANTUM THEORY
5. The Origin of the Old Quantum Theory 25
5a, The Postulates of Bohr 26
56. The WilsonSommerfeld Rules of Quantization 28
5c. Selection Rules. The Correspondence Principle 29
6. The Quantization of Simple Systems 30
6a. The Harmonic Oscillator. Degenerate States 30
66. The Rigid Rotator 31
6c. The Oscillating and Rotating Diatomic Molecule 32
6d. The Particle in a Box 33
6c. Diffraction by a Crystal Lattice 34
7. The Hydrogen Atom 36
7a. Solution of the Equations of Motion 36
76. Application of the Quantum Rules. The Energy Levels . . 39
7c. Description of the Orbits 43
7a*. Spatial Quantization 45
8. The Decline of the Old Quantum Theory 47
v
vi CONTENTS
Section Page
Chapter III
THE SCHRODINGER WAVE EQUATION WITH THE
HARMONIC OSCILLATOR AS AN EXAMPLE
9. The Schrodinger Wave Equation 50
9a. The Wave Equation Including the Time 53
96. The Amplitude Equation 56
9c. Wave Functions. Discrete and Continuous Sets of Charac
teristic Energy Values 58
9d. The Complex Conjugate Wave Function ty*(x,t) . . . . 63
10. The Physical Interpretation of the Wave Functions 63
10a. ty*(x,t)ty(x,t) as a Probability Distribution Function. . 63
106. Stationary States 64
10c. Further Physical Interpretation. Average Values of
Dynamical Quantities 65
11. The Harmonic Oscillator in Wave Mechanics . . 67
11a. Solution of the Wave Equation 67
116. The Wave Functions for the Harmonic Oscillator and their
Physical Interpretation ... 73
lie. Mathematical Properties of the Harmonic Oscillator Wave
Functions 77
Chapter IV
THE WAVE EQUATION FOR A SYSTEM OF POINT
PARTICLES IN THREE DIMENSIONS
12. The Wave Equation for a System of Point Particles 84
12a. The Wave Equation Including the Time 85
126. The Amplitude Equation 86
12c. The Complex Conjugate Wave Function ty*(xi • ■ • z#, 88
12d. The Physical Interpretation of the Wave Functions ... 88
13 The Free Particle 90
14. The Particle in a Box 95
15. The Threedimensional Harmonic Oscillator in Cartesian Coordi
nates 100
16. Curvilinear Coordinates 103
17. The Threedimensional Harmonic Oscillator in Cylindrical Coordi
nates 105
Chapter V
THE HYDROGEN ATOM
18. The Solution of the Wave Equation by the Polynomial Method
and the Determination of the Energy Levels 113
18a. The Separation of the Wave Equation. The Transla
tional Motion 113
186. The Solution of the <p Equation 117
CONTENTS Vit
Suction Page
18c. The Solution of the # Equation 118
ISd. The Solution of the r Equation 121
18c. The Energy Levels 124
19. Legendre Functions and Surface Harmonics 125
19a. The Legendre Functions or Legendre Polynomials 126
196. The Associated Legendre Functions 127
20. The Laguerre Polynomials and Associated Laguerre Functions . . 129
20a. The Laguerre Polynomials 129
206. The Associated Laguerre Polynomials and Functions ... 131
21. The Wave Functions for the Hydrogen Atom 132
21a. Hydrogenlike Wave Functions 132
216. The Normal State of the Hydrogen Atom 139
21c. Discussion of the Hydrogenlike Radial Wave Functions. . 142
21d. Discussion of the Dependence of the Wave Functions
on the Angles t? and <p 146
Chapter VI
PERTURBATION THEORY
22. Expansions in Series of Orthogonal Functions 151
23 ^Firstorder Perturbation Theory for a Nondegenerate Level . . . 156
23a. A Simple Example: XhaJPexluikejj, gax mnnin Qs rillator. . 160
9Zh An F, X arnp1p Tin^JsTnrmfll H'^"™ Atom 162
24. J£i£storder Perturbation Theory for a Degenerate Level 165
24a. An Example: Application of a Perturbation to a Hydrogen
Atom 172
25. Secondorder Perturbation Theory 176
25a. An Example: The Stark Effect of the Plane Rotator . . .177
Chapter VII
THE VARIATION METHOD AND OTHER APPROXIMATE
METHODS
26. The Variation Method 180
26a. The Variational Integral and its Pr operties 180
266. An Example: The Normal State of t he Helium A tom . . . 184
26c. Application of the Variation Method to Other StaTes ... 186
26d. Linear Variation Functions 186
26c. A More General Variation Method 189
27. Other Approximate Methods 191
27a. A Generalized Perturbation Theory 191
276. The WentzelKramersBrillouin Method 198
27c. Numerical Integration 201
27d. Approximation by the Use of Difference Equations .... 202
27c. An Approximate Secondorder Perturbation Treatment . . 204
Via CONTENTS
Suction Paqb
Chapter VIII
THE SPINNING ELECTRON AND THE PAULI EXCLUSION
PRINCIPLE, WITH A DISCUSSION OF THE HELIUM ATOM
28. The Spinning Electron 207
29. The Helium Atom. The Pauli Exclusion Principle 210
29a. The Configurations ls2s and ls2p 210
296. The Consideration of Electron Spin. The Pauli Exclusion •
Principle 214
29c. The Accurate Treatment of the Normal Helium Atom. . . 221
29d. Excited States of the Helium Atom 225
29e. The Polarizability of the Normal Helium Atom 226
Chapter IX
MANYELECTRON ATOMS
30. Slater's Treatment of Complex Atoms 230
30a. Exchange Degeneracy 230
306. Spatial Degeneracy 233
30c. Factorization and Solution of the Secular Equation. . . 235
30d. Evaluation of Integrals ... * 239
30e. Empirical Evaluation of Integrals. Applications. . . . 244
31. Variation Treatments for Simple Atoms 240
31a. The Lithium Atom and Threeelectron Ions . .... 247
316. Variation Treatments of Other Atoms. . . 249
32. The Method of the Selfconsistent Field 250
32a. Principle of the Method 250
326. Relation of the Selfconsistent Field Method to the Varia
tion Principle ^ . . . 252
32c. Results of the Selfconsistent Field Method 254
33. Other Methods for Manyelectron Atoms 256
33a. Semiempirical Sets of Screening Constants 256
336. The ThomasFermi Statistical Atom 257
Chapter X
THE ROTATION AND VIBRATION OF MOLECULES
34. The Separation of Electronic and Nuclear Motion 259
35. The Rotation and Vibration of Diatomic Molecules 263
35a. The Separation of Variables and Solution of the Angular
Equations 264
356. The Nature of the Electronic Energy Function 266
35c. A Simple Potential Function for Diatomic Molecules . . . 267
35d. A More Accurate Treatment. The Morse Function . . . 271
36. The Rotation of Polyatomic Molecules 275
36a. The Rotation of Symmetricaltop Molecules 275
366. The Rotation of Unsymmetricaltop Molecules 280
CONTENTS ix
Section Page
37. The Vibiation of Polyatomic Molecules 282
37a. Normal Coordinates in Classical Mechanics 282
376. Normal Coordinates in Quantum Mechanics 288
38. The Rotation of Molecules in Crystals 290
Chapter XI
PERTURBATION THEORY INVOLVING THE TIME, THE
EMISSION AND ABSORPTION OF RADIATION, AND THE
RESONANCE PHENOMENON
39. The Treatment of a Timedependent Perturbation by the Methoa
of Variation of Constants .... 294
39a. A Simple Example .... 296
40. The Emission and Absorption of Radiation. V .... 299
40a. The Einstein Transition Probabilities .... 299
406. The Calculation of the Einstein Transition Probabilities
by Perturbation Theory 302
40c. Selection Rules and Intensities for the Harmonic Oscillator 306
40d. Selection Rules and Intensities for Surface harmonic Wave
Functions 306
40e. Selection Rules and Intensities for the Diatomic Moleculev.
The FranckCondon Principle 309
40/. Selection Rules and Intensities for the Hydrogen Atom . .312
40a. Even and Odd Electronic States and their Selection Rules. 313
41. The Resonance Phenomenon 314
41a. Resonance in Classical Mechanics 315
416. Resonance in Quantum Mechanics 318
41c. A Further Discussion of Resonance 322
Chapter XII
THE STRUCTURE OF SIMPLE MOLECULES
42. The Hydrogen Moleculeion 327
42a. A Very Simple Discussion 327
426. Other Simple Variation Treatments 331
42c. The Separation and Solution of the Wave Equation .... 333
42a\ Excited States of the Hydrogen Moleculeion 340
43. The Hydrogen Molecule 340
43a. The Treatment of Heitler and London. 340
436. Other Simple Variation Treatments . . . 345
43c. The Treatment of James and Coolidge 349
43d. Comparison with Experiment 351
43e. Excited States of the Hydrogen Molecule 353
43/. Oscillation and Rotation of the Molecule. Ortho and
Para Hydrogen 355
44. The Helium Moleculeion He£ and the Interaction of Two Normal
Helium Atoms 358
44a. The Helium Moleculeion Hef 358
INTRODUCTION TO QUANTUM
MECHANICS
CHAPTER I
SURVEY OF CLASSICAL MECHANICS
The subject of quantum mechanics constitutes the most recent
step in the very old search for the general laws : jroyjyming the
motion of matter. For a long time investigators confined their
efforts to studying the dynamics of bodies of macroscopic dimen
sions, and while the science of mechanics remained in that
stage it was properly considered a branch of physics. Since
the development of atomic theory there has been a change of
emphasis. It was recognized that the older laws are not correct
when applied to atoms and electrons, without considerable
modification. Moreover, the success which has been obtained
in making the necessary modifications of the older laws has also
had the result of depriving physics of sole claim upon them, since
it is now realized that the combining power of atoms and, in
fact, all the chemical properties of atoms and molecules are
explicable in terms of the laws governing the motions of the
electrons and nuclei composing them.
Although it is the modern theory of quantum mechanics in
which we are primarily interested because of its applicatiqnsjto
chemical problems, it is desirable for us first to discuss briefly
the background of classical mechanics from which it was devel
oped. By so doing we not only follow to a certain extent the
historical development, but we also introduce in a more familiar
form many concepts which are retained in the later theory. We
shall also treat certain problems in the first few chapters by the
methods of the older theories in preparation for their later treat
ment by quantum mechanics. It is for this reason that the
student is advised to consider the exercises of the first few
chapters carefully and to retain for later reference the results
which are secured.
1
2 SURVEY OF CLASSICAL MECHANICS [11
In the first chapter no attempt will be made to give any parts
of classical dynamics but those which are useful in the treatment
of atomic and molecular problems. With this restriction, we
have felt justified in omitting discussion of the dynamics of rigid
bodies, nonconservative systems, nonholonomic systems, sys
tems involving impact, etc. Moreover, no use is made of
Hamilton's principle or of the HamiltonJacobi partial differential
equation. By thus limiting the subjects to be discussed, it is
possible to give in a short chapter a thorough treatment of
Newtonian systems of point particles.
1. NEWTON'S EQUATIONS OF MOTION IN THE LAGRANGIAN
FORM
The earliest formulation of dynamical laws of wide application
is that of Sir Isaac Newton. If we adopt the notation Xi, y t , Zi
for the three Cartesian coordinates of the ith particle with
mass Wi, Newton's equations for n point particles are
m x Xi = X t) )
m x iji = Y if } i = 1, 2, • • • , n, (11)
m x yi = i if /
miii = Z X) )
where X», Y x , Z t are the three components of the force acting on
the ^th particle. There is a set of such equations for each
particle. Dots refer to differentiation with respect to time, so
that
d 2 r
*<  w ™
By introducing certain familiar definitions we change Equation
11 into a form which will be more useful later. We define as
the kinetic energy T (for Cartesian coordinates) the quantity
T = Mm x {±\ + y\ + i\) + • • • + V 2 m n {xl + yl + zl)
n
= h X mii± < + y? + #• ( 1_3 )
If we limit ourselves to a certain class of systems, called conserva
tive systems, it is possible to define another quantity, the potential
energy V, which is a function of the coordinates Xii/iZi • • •
x n y n z n of all the particles, such that the force components acting
11] NEWTON'S EQUATIONS OF MOTION 3
on each particle are equal to partial derivatives of the potential
energy with respect to the coordinates of the particle (with
negative sign) ; that is,
A{ = —  — >
Y< = —,) i = 1, 2, • • • , n. (14)
It is possible to find a function V which will express in this manner
forces of the types usually designated as mechanical, electrostatic,
and gravitational. Since other types of forces (such as electro
magnetic) for which such a potentialenergy function cannot
be set up are not important in chemical applications, we shall
not consider them in detail.
With these definitions, Newton's equations become
(l5a)
(156)
(l5c)
There are three such equations for every particle, as before.
These results are definitely restricted to Cartesian coordinates;
but by introducing a new function, the Lagrangian function L f
defined for Newtonian systems as the difference of the kinetic
and potential energy,
L = L(xi, y h Zi, • • • , x n , y n , z n , x h • • • , z n ) =
T  V, (16)
we can throw the equations of motion into a form which we shall
later prove to be valid in any system of coordinates (Sec. lc).
In Cartesian coordinates T is a function of the velocities
xi 9 • • • , i n only, and for the systems to which our treatment
is restricted V is a function of the coordinates only; hence the
equations of motion given in Equation 15 on introduction of
the function L assume the form
d^dT dV _
dt d±i dZi
= 0,
d dT dV _
dt dy { dy {
d dT dV
dt dii dZi
= 0,
= 0.
SURVEY OF CLASSICAL MECHANICS
d dL _
dt dxi
dL
dXi
= o,\
d dL _
eft d^t
d dL __
dt dii
dL
dyi
dL
dZi
= 0,1
i = 1, 2, • • • , 7
[Ila
(17)
In the following paragraphs a simple dynamical system is
discussed by the use of these equations.
la. The Threedimensional Isotropic Harmonic Oscillator. —
As an illustration of the use of the equations of motion in this
form, we choose a system which has played a very important
part in the development of quantum theory. This is the
farngnij^osctJjMor, a particle bound to an equilibrium position by
a force which increases in magnitude linearly with its distance
r from the point. In the threedimensional isotropic harmonic
oscillator this corresponds to a potential function }ikr 2 , represent
ing a force of magnitude kr acting in a negative direction; i.e.,
from the position of the particle to the origin, k is called the
force constant or Hooke'slaw constant. Using Cartesian coordi
nates we have
L = y 2 m(x> + y 2 + * 2 )  V2k(x> + y 2 + z 2 ), (18)
whence
rXmx) + kx = mx + kx = 0,]
my + ky =0,/ (1~9)
m'z + kz = O.j
Multiplication of the first member of Equation 19 by x gives
mi tt   kx % (1_10)
or
2 m dt 2 k dt ' U ll)
which integrates directly to
%mx 2 = Y 2 kx 2 + constant. (112)
The constant of integration is conveniently expressed as x /ikx\.
Ila] NEWTON'S EQUATIONS OF MOTION
Hence
dx ___ fie
dl " \m
or
and similarly
(*l  * 2 ), (113)
or, on introducing the expression \K 2 mv\ in place of the force
constant k y
which on integration becomes
2wvot + 5 X = sin 1 —
x = x sin (2irM + $*), (114)
2/ = t/o sin (aryrf + & y )\ (115)
2 = Zq sin (2tt^ < + 5.) /
In these expressions x , y , z , 8 X , 8 y , and 8 Z are constants of
integration, the values of which determine the motion in any
given case. The quantity v is related to the constant of the
restoring force by the equation
4r*mv$ = Jfc, (116)
so that the potential energy may be written as
V = 2^mv'ir\ (117)
As shown by the equations for x, y, and z, p is the frequency of
the motion. It is seen that the particle may be described as
carrying out independent harmonic oscillations along the x, y }
and z axes, with different amplitudes Xo, t/o, and Zq and different
phase angles 8 X , b yy and 8 Z , respectively.
The energy of the system is the sum of the kinetic energy and
the potential energy, and is thus equal to
}im(x 2 + y 2 + z 2 ) + 2jr 2 mvl(x 2 + y 2 + z 2 ).
On evaluation, it is found to be independent of the time, with the
value 2Tr 2 mvl(xl + y\ + zj) determined by the amplitudes of
oscillation.
The onedimensional harmonic oscillator, restricted to motion
along the x axis in accordance with the potential function
V = }ikx 2 = 2w 2 mv%x 2 , is seen to carry out harmonic oscillations
6 SURVEY OF CLASSICAL MECHANICS [Ilb
along this axis as described by Equation 114. Its total energy
is given by the expression 2w 2 mvlxl
lb. Generalized Coordinates. — Instead of Cartesian coordi
nates xi, 2/1, 2i, • • • , x n , y n , z n , it is frequently more convenient
to use some other «et of coordinates to specify the configuration
of the system. For example, the isotropic spatial harmonic
oscillator already discussed might equally well be described using
polar coordinates; again, the treatment of a system composed of
two attracting particles in space, which will be considered
later, would be very cumbersome if it were necessary to use
rectangular coordinates.
If we choose, any set of 3n coordinates, which we shall always
assume to be independent and at the same time sufficient in
number to specify completely the positions of the particles of
the system, then there will in general exist 3n equations, called
the equations of transformation, relating the new coordinates
qn to the set of Cartesian coordinates x t , y tf z i}
(118)
There is such a set of three equations for each particle i. The
functions / t , g i} hi may be functions of any or all of the Zn new
coordinates q k , so that these new variables do not necessarily
split into sets which belong to particular particles. For example,
in the case of two particles the six new coordinates may be the
three Cartesian coordinates of the center of mass together
with the polar coordinates of one particle referred to the other
particle as origin.
As is known from the theory of partial differentiation, it is
possible to transform derivatives from one set of independent
variables to another, an example of this process being
dt dq t dt "*" dq 2 dt "*" "*" dq 3n IF' ^~ iya ^
This same equation can be put in the much more compact form
3n
**«* (1196)
*i = /i(<?l> 02, ' *
• > 03«),
Vi = Vifal, 02, ' '
• , ?3n),
Zi = hi(q ly q 2 , ' '
• , Q*n).
XT' dZi .
Ilc] NEWTON'S EQUATIONS OF MOTION 7
This gives the relation between any Cartesian component of
velocity and the time derivatives of the new coordinates. Similar
relations, of course, hold for t/» and Zi for any particle. The
quantities % by analogy with £», are called generalized velocities,
even though they do not necessarily have the dimensions of
length divided by time (for example, qj may be an angle).
Since partial derivatives transform in just the same manner,
we have
_£Z" = _^Z^ _ ^Z d l} _ . . . _ ^z^?
dqj dxi dqj dyi dqj dz n dq s
= " 2(a*, Jg, + tyi "4/ + a£ wi) = Qi ' (1_20)
Since Q 3 is given by an expression in terms of V and qj which is
analogous to that for the force Xi in terms of V and x i} it is called
a generalized force.
In exactly similar fashion, we have
*! = ^2(*! d Ai + wdj A wdzA ( .
dq 3  2Z\d±i dqj "*" dy { dqj "*" di< dqj)' }
lc. The Invariance of the Equations of Motion in the Lagran
gian Form. — We are now in a position to show that when New
ton's equations are written in the form given by Equation 17
they are valid for any choice of coordinate system. For this
proof we shall apply a transformation of coordinates to Equa
tions 15, using the methods of the previous section. Multiplica
dx ' du '
tion of Equation l5a by — *> of 156 by ~) etc., gives
dqj dqj
dxiddT^ ^Z^i ^ ^
dqj dt d±i dxi dqj '
dzidW dVdXi = 0(
dqj dt d± 2 dx 2 dqj '} (122)
dXnddT^ dVdXn = Q 
dqj dt dx n dx n dq, * *
with similar equations in y and z. Adding all of these together
gives
8 SURVEY OF CLASSICAL MECHANICS [Ilc
2l \dqj dt d±i "*" dqi dt dj/i ^ dq, dt diij "*" dq, ' K }
where the result of Equation 120 has been used. In order to
reduce the first sum, we note the following identity, obtained by
differentiating a product,
dXid/dT\ = d/dTdxA _ dTd/dxA
dqjdtydXi) dt\dXidqi) dXidt\dqj)' ( )
From Equation 11% we obtain directly
ft = 1/ «**>
Furthermore, because the order of differentiation is immaterial,
we see that
d/dXi\ = ^±_/<tei\ d = N^A/^A
cttUfc/ <Zl dq k \Sqi ) Qk 2j dq\dq k ) qk
& = 1 *l
By introducing Equations 126 and 125 in 124 and using the
result in Equation 123, we get
dTdyi
dy { dqj
2(d/dT d±i , dTdyi dT diA __ /dT dxt
\dt\dXi dqj + dyi dqj + di; dqj) \dxi dqj
which, in view of the results of the last section, reduces to
ddT dT dV_
Finally, the introduction of the Lagrangian function L = T — V,
with V a function of the coordinates only, gives the more compact
form
Ild] NEWTON'S EQUATIONS OF MOTION 9
(It is important to note that L must be expressed as a function
of the coordinates and their first timederivatives.)
Since the above derivation could be carried out for any value
of j, there are 3n such equations, one for each coordinate q,.
They are called the equations of motion in the Lagrangian form
and are of great importance. The method by which they were
derived shows that they are independent of the coordinate
system.
We have so far rather limited the types of systems considered,
but Lagrange's equations are much more general than we have
indicated and by a proper choice of the function L nearly all dynami
cal problems can be treated with their use. These equations are
therefore frequently chosen as the fundamental postulates of
classical mechanics instead of Newton's laws.
Id. An Example : The Isotropic Harmonic Oscillator in Polar
Coordinates. — The example which we have treated in Section la
can equally well be solved by the use of polar coordinates r,
#, and <p (Fig. 11) . The equations of transformation correspond
ing to Equation 118 are
x = rsin#cos^,)
y = r sin # sin tp, \ (130)
z = rcostf. j
With the use of these we find for the kinetic and potential energies
of the isotropic harmonic oscillator the following expressions:
T = ~ m(x 2 + y 2 + z 2 ) =  (r 2 + rW + r 2 sin 2 * <p 2 )) (1 _ 31)
V = 27r 2 mv 2 r 2 , j
and
L = T  V = ~(r 2 + r 2 & 2 + r 2 sin 2 #<p 2 )  2nr 2 mv\r 2 . (132)
The equations of motion are
£ $k _ M = ^(mr 2 ^)  mr 2 sin a cos &<p 2 = 0, (134)
at $# oft at
t*z. :r = T(wr) — mr# 2 — mr sin 2 &v> 2 + Ar 2 mvlr «* 0.
dtdr dr dC ' °
(135)
10
SURVEY OF CLASSICAL MECHANICS
Iid
In Appendix II it is shown that the motion takes place in a
plane containing the origin. This conclusion enables us to
simplify the problem by making a change of variables. Let us
introduce new polar coordinates r, #', x such that at the time
t = the plane determined by the vectors r and v, the position
and velocity vectors of the particle at t = 0, is normal to the new
z' axis. This transformation is known in terms of the old set of
coordinates if two parameters # and v? , determining the position
of the axis z' in terms of the old coordinates, are given (Fig. 12).
Fig. 11. — The relation of polar coor
dinates r, t? , and tp to Cartesian axes.
Fig. 12. — The rotation of axes.
In terms of the new coordinates, the Lagrangian function L
and the equations of motion have the same form as previously,
because the first choice of axis direction was quite arbitrary.
However, since the coordinates have been chosen so that the
plane of the motion is the x'y' plane, the angle &' is always equal
to a constant, v/2. Inserting this value of & in Equation 133
and writing it in terms of x instead of <p, we obtain
dt
(mr 2 x) = 0,
which has the solution
mr 2 \ = p X) a constant.
The r equation, Equation 135, becomes
n(™>r) — mry} + 4r 2 mvlr = 0,
(136)
(137)
Me] NEWTON'S EQUATIONS OF MOTION 11
or, using Equation 137,
J t (mr) ~£i + 4r*m,tfr = 0, (138)
an equation differing from the related onedimensional Cartesian
coordinate equation by the additional term — p$/mr z which
represents the centrifugal force.
Multiplication by f and integration with respect to the time
gives
f2 = ^ ~ 47r2 "° r2 + b > (1 " 39)
so that r = ( — ^ 9  \>w 2 v 2 r 2 + b) •
\ m 2 r 2 u /
This can be again integrated, to give
t  U
P rdr
2J (a
dx
+ bx + cx 2 Y
in which x = r 2 , a = — p/m 2 , b is the constant of integration in
Equation 139, and c = —iw 2 vl. This is a standard integral
which yields the equation
g— 2 {& + A sin 4irv (t  to)},
with A given by
A
= ^ 2 "
16?r 2 vgp 2
m L
We have thus obtained the dependence of r on the time, and
by integrating Equation 137 we could obtain xasa function of
the time, completing the solution. Elimination of the time
between these two results would give the equation of the orbit,
which is an ellipse with center at the origin. It is seen that the
constant v again occurs as the frequency of the motion.
le. The Conservation of Angular Momentum. — The example
worked out in the previous section illustrates an important
principle of wide applicability, the principle of the conservation
of angular momentum.
12
SURVEY OF CLASSICAL MECHANICS
Ule
Equation 137 shows that when x is the angular velocity of the
particle about a fixed axis z' and r is the distance of the particle
from the axis, the quantity p x = mr 2 \ is a constant of the motion. *
This quantity is called the angular momentum of the particle
about the axis z'.
It is not necessary to choose an axis normal to the plane of the
motion, as z' in this example, in order to apply the theorem.
Thus Equation 133, written for arbitrary direction z, is at once
integrable to
rar 2 sin 2 &<p = p^ a constant. (140)
Here r sin & is the distance of the particle from the axis z, so that
the left side of this equation is the angular momentum about the
axis z. 2 It is seen to be equal to a constant, p*.
Fig. 13. — Figure showing the relation between dx, d&, and d<p.
In order to apply the principle, it is essential that the axis of
reference be a fixed axis. Thus the angle & of polar coordinates
has associated with it an angular momentum p# = mrH about
an axis in the xy plane, but the principle of conservation of
angular momentum cannot be applied directly to this quantity
because the axis is not, in general, fixed but varies with ^. A
simple relation involving p* connects the angular momenta
1 The phrase a constant of the motion is often used in referring to a constant
of integration of the equations of motion for a dynamical system.
2 This is sometimes referred to as the component of angular momentum
along the axis z.
Ile] NEWTON'S EQUATIONS OF MOTION 13
p x and p<p about different fixed axes, one of which, p x , relates
to the axis normal to the plane of the motion. This is
p x dx = P*d& + pyd<p, (141)
an equation easily derived by considering Figure 13. The
sides of the small triangle have the lengths r sin dd(p, rdx, and
rd&. Since they form a right triangle, these distances are
connected by the relation
r\d%Y = r 2 sin 2 #(d<p) 2 + r 2 (d#) 2 ,
which gives, on introduction of the angular velocities x, <P, and #
and multiplication by m/dt,
mr 2 xdx = mr 2 sin 2 &<pd<p + mr 2 $d&.
Equation 141 follows from this and the definitions of p X} p*,
and p v .
Conservation of angular momentum may be applied to more
general systems than the one described here. It is at once
evident that we have not used the special form of the potential
energy expression except for the fact that it is independent of
direction, since this function enters into the r equation only.
Therefore the above results are true for a particle moving in
any spherically symmetric potential field.
Furthermore, we can extend the theorem to a collection of
point particles interacting with each other in any desired way
but influenced by external forces only through a spherically
symmetric potential function. If we describe such a system by
using the polar coordinates of each particle, the Lagrangian
function is
n
L = V 2 X m ^ + r & + r $ sin2 *&f)  V. (142)
i = i
Instead of <pi, <p 2 , • • • , <p n) we now introduce new angular
coordinates a, 0, • • • , k given by the linear equations
<Pi = a + bip +  • • + kiK 9 \
<p 2 = a + b 2 p + ' ' ' + k 2 K,l ,. . ox
> (143)
>\
<Pn = a +bnP + * ' * +k n K.)
The vaiues given the constants b u • • • , k n are unimportant so
long as they make the above set of equations mutually independ
14 SURVEY OF CLASSICAL MECHANICS [I2a
ent. a is an angle about the axis z such that if a is increased
by Aa, holding /?, • * • , k constant, the effect is to increase each
ipi by Aa, or, in other words, to rotate the whole system of particles
about z without changing their mutual positions. By hypothesis
the value of V is not changed by such a rotation, so that V is
independent of a. We therefore obtain the equation
daL_6L = d_3T =
dt da da dt da U **'
Moreover, from Equation 142 we derive the relation
2Sr£ 2'* •*■» < i «>
dT
da
t = l
Hence, calling the distance r t sin d x of the tth particle from the
z axis pi, we obtain the equation
n
^m i p^<p l = constant. (146)
i = i
This is the more general expression of the principle of the con
servation of angular momentum which we were seeking. In
such a system of many particles with mutual interactions, as,
for example, an atom consisting of a number of electrons and a
nucleus, the individual particles do not in general conserve
angular momentum but the aggregate does.
The potentialenergy function V need be only cylindrically
symmetric about the axis z for the above proof to apply,
since the essential feature was the independence of V on the angle
a about z. However, in that case z is restricted to a particular
direction in space, whereas if V is spherically symmetric the
theorem holds for any choice of axis.
Angular momenta transform like vectors, the directions of the
vectors being the directions of the axes about which the angular
momenta are determined. It is customary to take the sense
of the vectors such as to correspond to the righthand screw rule.
3. THE EQUATIONS OF MOTION IN THE HAMILTONIAN FORM
2a. Generalized Momenta. — In Cartesian coordinates the
momentum related to the direction x k is m k x kj which, since V is
I2a] EQUATIONS OF MOTION IN HAMILTONIAN FORM 15
restricted to be a function of the coordinates only, can be written
as
^ = 1 = 1' * = l,2,,3n. (21)
Angular momenta can likewise be expressed in this manner.
Thus, for one particle in a spherically symmetric potential field,
the angular momentum about the z axis was defined in Section le
by the expression
ZV = mp 2 <p = mr 2 sin 2 &<p. (22)
Reference to Equation 131, which gives the expression for the
kinetic energy in polar coordinates, shows that
*  *  w (2 ~ 3)
Likewise, in the case of a number of particles, the angular
momentum conjugate to the coordinate a is
as shown by the discussion of Equation 146. By extending
this to other coordinate systems, the generalized momentum pk
conjugate to the coordinate q k is defined as
Vk = Jg, k = 1, 2, • • • , 3n. (25)
The form taken by Lagrange's equations (Eq. 129) when the
definition of p* is introduced is
p k = ^, k = 1, 2, • • • , 3n, (26)
so that Equations 25 and 26 form a set of 6n firstorder dif
ferential equations equivalent to the 3n secondorder equations
of Equation 129.
— being in general a function of both the q'a and q's, the
oqk
definition of p k given by Equation 25 provides 3n relations
between the variables q k) <ik, and pk, permitting the elimination
of the 3n velocities q k , so that the system can now be described
in terms of the 3n coordinates qk and the 3n conjugate momenta
16 SURVEY OF CLASSICAL MECHANICS (I2c
p k . Hamilton in 1834 showed that the equations of motion can
in this way be thrown into an especially simple form, involving
a function H of the p k 's and q k s called the Hamiltonian function.
2b. The Hamiltonian Function and Equations. — For con
servative systems. 1 we shall show that the function H is the total
energy (kinetic plus potential) of the system, expressed in terms
of the pk's and q k $. In order to have a definition which holds
for more general systems, we introduce H by the relation
3n
H = ]£?*&  L(q k , q k ). (27)
Although this definition involves the velocities q k) H may be made
a function of the coordinates and momenta only, by eliminating
the velocities through the use of Equation 25. From the
definition we obtain for the total differential of H the equation
3n 3n 3n 3n
dH = ^Pkdq k + 2«*dp*  2jdq~ k dqk " 2^^*' ^ 2 " 8 ^
or, using the expressions for p k and p k given in Equations 25 and
26 (equivalent to Lagrange's equations),
3n
dH = 2)(g*dpib  Pkdqk), (29)
fc = l
whence, if // is regarded as a function of the q k s and p k s, we
obtain the equations
dH _
dpk
4k.
>
dH _
dq k
—
Pk,
1, 2, • • • , 3n. (210)
These are the equations of motion in the Hamiltonian or canonical
form. ~ K
2c. The Hamiltonian Function and the 'Energy. — Let us con
sider the time dependence of H for a conservative system. We
have
1 A conservative system is a system for which H does not depend explicitly
on the time t. We have restricted our discussion to conservative systems by
assuming that the potential function V does not depend on t.
181 THE EMISSION AND ABSORPTION OF RADIATION 21
coordinate but only its derivative. Such a coordinate is cahea a cyclic
coordinate.
3. THE EMISSION AND ABSORPTION OF RADIATION
The classical laws of mechanical and electromagnetic theory
permit the complete discussion of the emission and absorption of
electromagnetic radiation by a system of electrically charged
particles. In the following paragraphs we shall outline the
results of this discussion. It is found that these results are not
in agreement with experiments involving atoms and molecules;
it was, indeed, just this disagreement which was the principal
factor in leading to the development of the Bohr theory of the
atom and later of the quantum mechanics. Even at the present
time, when an apparently satisfactory theoretical treatment of
dynamical systems composed of electrons and nuclei is provided
by the quantum mechanics, the problem of the. emission and
absorption of radiation still lacks a satisfactory solution, despite
the concentration of attention on it by the most able theoretical
physicists. It will be shown in a subsequent chapter howeyer,
that, despite our lack of a satisfactory conception of the nature
of electromagnetic radiation, equations similar to the classical
equations of this section can be formulated which represent
correctly the emission and absorption of radiation by' atomic
systems to within the limits of error of experiment.
According to the classical theory the rate of emission of radiant
energy by an accelerated particle of electric charge e is
dt 3c 3 '
dE
in which — ^ is the rate at which the energy E of the particle
is converted into radiant energy, v is the acceleration of the
particle, and c the velocity of light.
Let us first consider a system of a special type, in which a
particle of charge e carries out simple harmonic oscillation
with frequency v along the x axis, according to the equation
x = x cos 2rvt. (32)
Differentiating this expression, assuming that z* is independent
of the time, we obtain for the acceleration the value
v ss x == — 4irV£o cos %cvt.
22 SURVEY OF CLASSICAL MECHANICS [I»
The average rate of emission of radiant energy by such a system
is consequently
~~dt = 3c* ' (3 ^ }
inasmuch as the average value cos 2 2rvt over a cycle is "onehalf.
As a result of the emission of energy, the amplitude x of the
motion will decrease with time; if the fractional change in
energy during a cycle of the motion is small, however, this equa
tion retains its validity.
The radiation emitted by such a system has the frequency v of the
emitting system. It is planepolarized, the plane of the electric
vector being the plane which includes the x axis and the direction
of propagation of the light.
In case that the particle carries out harmonic oscillations along
all three axes x, y, and z, with frequencies v x> v Vt and v z and
amplitudes (at a given time) Xo, yo, and z 0f respectively, the total
rate of emission of radiant energy will be given as the sum of
three terms similar to the right side of Equation 34, one giving
the rate of emission of energy as light of frequency v Xl one of
v V) and one of v t .
If the motion of the particle is not simple harmonic, it can be
represented by a Fourier series or Fourier integral as a sum or
integral of harmonic terms similar to that of Equation 32;
light of frequency characteristic of each of these terms will then
be emitted at a rate given by Equation 34, the coefficient of the
Fourier term being introduced in place of x .
The emission of light by a system composed of several inter
acting electrically charged particles is conveniently discussed in
the following way. A Fourier analysis is first made of the
motion of the system in a given state to resolve it into harmonic
terms. For a given term, corresponding to a given frequency
of motion v y the coefficient resulting from the analysis (which is a
function of the coordinates of the particles) is expanded as a
power series in the quantities Xi/\ • • • , z n /\ in which xi,
• • • , z n are the coordinates of the particles relative to some
origin (such as the center of mass) and X = c/v is the wave length
of the radiation with frequency v. The term of zero degree in
this expansion is zero, inasmuch as the electric charge of the
system does not change with time. The term of first degree
involves, in addition to the harmonic function of the time, only
14] SUMMARY OF CHAPTER I 23
a function of the coordinates. The aggregate of these first
degree terms in the coordinates with their associated time factors,
summed over all frequency values occurring in the original
Fourier analysis, represents a dynamical quantity known as the
electric moment of the system, a vector quantity P defined as
P = 2)*r< f (35)
i
in which r» denotes the vector from the origin to the position of
the ith. particle, with charge e». Consequently to this degree of
approximation the radiation emitted by a system of several
particles can be discussed by making a Fourier analysis of the
electric moment P. Corresponding to each term of frequency v
in this representation of P, there will be emitted radiation of
frequency v at a rate given by an equation similar to Equation
34, with exo replaced by the Fourier coefficient in the electric
moment expansion. The emission of radiation by this mechanism
is usually called dipole emission, the radiation itself sometimes
being described as dipole radiation.
The quadratic terms in the expansions in powers of Xi/\
• • • , Zn/X form a quantity Q called the quadrupole mo'ment
of the system, and higher powers form higher moments. The rate
of emission of radiant energy as a result of the change of quadru
pole and higher moments of an atom or molecule is usually
negligibly small in comparison with the rate of dipole emission,
and in consequence dipole radiation alone is ordinarily discussed.
Under some circumstances, however, as when the intensity of
dipole radiation is zero and the presence of very weak radiation
can be detected, the process of quadrupole emission is important.
4. SUMMARY OF CHAPTER I
The purpose of this survey of classical mechanics is twofold:
first, to indicate the path whereby the more general formulations
of classical dynamics, such as the equations of motion of Lagrange
and of Hamilton, have been developed from the original equations
of Newton; and second, to illustrate the application of these
methods to problems which are later discussed by quantum
mechanical methods.
In carrying out the first purpose, we have discussed Newton's
equations in Cartesian coordinates and then altered their form by
24 SURVEY OF CLASSICAL MECHANICS [M
the introduction of the kinetic and potential energies. By
defining the Lagrangian function for the special case of Newtonian
systems and introducing it into the equations of motion, Newton's
equations were then thrown into the Lagrangian form. Follow
ing an introductory discussion of generalized coordinates, the
proof of the general validity of the equations of motion in the
Lagrangian form for any system of coordinates has been given;
and it has also been pointed out that the Lagrangian form
of the equations of motion, although we have derived it from the
equations of Newton, is really more widely applicable than
Newton's postulates, because by making a suitable choice of the
Lagrangian function a very wide range of problems can be
treated in this way.
In the second section there has been derived a third form for
the equations of motion, the Hamiltonian form, following the
introduction of the concept of generalized momenta, and the rela
tion between the Hamiltonian function and the energy has been
discussed.
In Section 3 a very brief discussion of the classical theory of
the radiation of energy from accelerated charged particles has
been given, in order to have a foundation for later discussions
of this topic. Mention is made of both dipole and quadrupole
radiation.
Finally, several examples (which are later solved by the use of
quantum mechanics), including the threedimensional harmonic
oscillator in Cartesian and in polar coordinates, have been
treated by the methods discussed in this chapter.
General References on Classical Mechanics
W. D. MacMillan: " Theoretical Mechanics. Statics and the Dynamics
of a Particle," McGrawHill Book Company, Inc., New York, 1932.
S. L. Loney: "Dynamics of a Particle and of Rigid Bodies," Cambridge
University Press, Cambridge, 1923..
J. H. Jeans: "Theoretical Mechanics," Ginn and Company, Boston, 1907.
E. T. Whittaker: "Analytical Dynamics," Cambridge University Press,
Cambridge, 1928.
R. C. Tolman: "Statistical Mechanics with Applications to Physics and
Chemistry," Chemical Catalog Company, Inc., New York, 1927, Chap. II,
The Elements of Classical Mechanics.
W. E. Byerly: "Generalized Coordinates," Ginn and Company, Boston,
1916.
CHAPTER II
THE OLD QUANTUM THEORY
5. THE ORIGIN OF THE OLD QUANTUM THEORY
The old quantum theory was born in 1900, when Max Planck 1
announced his theoretical derivation of th& .distribution. law for,
blackbody radiation which he had previously formulated from
empirical considerations. He showed that the results of experi
ment on the distribution of energy with frequency of radiation
in equilibrium with matter at a given temperature can be
accounted for by postulating that the vibrating particles of
matter (considered to act as harmonic oscillators) do not emit
or absorb light continuously but instead only in discrete quanti
ties of magnitude hv proportional to the frequency v of the light.
The constant of proportionality, h, is a new constant of nature;
it is called Planck 1 s constant and has the magnitude 6.547 X 10~ 27
erg sec. Its dimensions (energy X time) are those of the old
dynamical quantity called action; they are such that the product
of h and frequency v (with dimensions sec 1 ) has the dimensions
of energy. The dimensions of h are also those of angular momen
tum, and we shall see later that just as hv is a quantum of radiant
energy of frequency v y so is h/2ic a natural unit or quantum of
angular momentum.
The development of the quantum theory was at first slow. It
was not until 1905 that Einstein 2 suggested that the quantity
of radiant energy hv was sent out in the process of emission of
light not in all directions but instead unidirectionally, like a
particle. The name light quantum or photon is applied to such a
portion of radiant energy. Einstein also discussed the photo
electric effect, the fundamental processes of photochemistry,
and the heat capacities of solid bodies in terms of 'the quantum
theory. When light falls on a metal plate, electrons are emitted
from it. The maximum speed of these photoelectrons, however,
1 M. Planck, Ann. d. Phys. (4) 4, 553 (1901).
2 A. Einstein, Ann. d. Phys. (4) 17, 132 (1905).
25
26 THE OLD QUANTUM THEORY II5a
is not dependent on the intensity of the light, as would be
expected from classical electromagnetic theory, but only on its
frequency; Einstein pointed out that this is to be expected from
the quantum theory, the process of photoelectric emission involv
ing the conversion of the energy hv of one photon into the kinetic
energy of a photoelectron (plus the energy required to remove
the electron from the metal). Similarly, Einstein's law of
photochemical equivalence states that one molecule may be
activated to chemical reaction by the absorption of one photon.
The third application, to the heat capacities of solid bodies,
marked the beginning of the quantum theory of material systems.
Planck's postulate regarding the emission and absorption of
radiation in quanta hv suggested that a dynamical system such
as an atom oscillating about an equilibrium position with fre
quency vo might not be able to oscillate with arbitrary energy,
but only with energy values which differ from one another by
integral multiples of hv . From this assumption and a simple
extension of the principles of statistical mechanics it can be
shown that the heat capacity of a solid aggregate of particles
should not remain constant with decreasing temperature, but
should at some low temperature fall off rapidly toward zero.
This prediction of Einstein, supported by the earlier experi
mental work of Dewar on diamond, was immediately verified
by the experiments of Nernst and Eucken on various substances;
and quantitative agreement between theory and experiment for
simple crystals was achieved through Debye's brilliant refinement
of the theory. 1
6a. The Postulates of Bohr. — The quantum theory had
developed to this stage before it became possible to apply it
to the hydrogen atom; for it was not until 1911 that there
occurred the discovery by Rutherford of the nuclear constitu
tion of the atom — its composition from a small heavy posi
tively charged nucleus and one or more extranuclear electrons.
Attempts were made immediately to apply the quantum theory to
the hydrogen atom. The successful effort of Bohr 2 in 1913,
despite its simplicity, may well be considered the greatest single
step in the development of the theory of atomic structure.
1 P. Debyb, Ann. d. Phya. (4) 39, 789 (1912); see also M. Born and T. votf
KXrmAn, Phya. Z. 13, 297 (1912); 14, 15 (1913).
S XT T>^ CT , DL.'I %* A« t /*r\*n\
H5a] THE ORIGIN OF THE OLD QUANTUM THEORY 27
It was clearly evident that the laws of classical mechanical and
electromagnetic theory could not apply to the Rutherford
hydrogen atom. According to classical theory the electron
in a hydrogen atom, attracted toward the nucleus by an inverse
square Coulomb force, would describe an elliptical or circular
orbit about it, similar to that of the earth about the sun. ^The
a ccelerati o n of the ch ar^edjgartjcles would lead to the emission
of light, with frequencies equal to the mechanical frequency
of the electron in its orbit, and to multiples of this as overtones.
With the emission of energy, the radius of the orbit_ would
diminish and the mechanical frequency would change. Hence
the emitted light should show a wide range of frequencies.) This
is not at all what is observed — the radiation emitted by hydrogen
atoms is confined to spectral lines of sharply defined frequencies,
and, moreover, these frequencies are not related to one another
by integral factors, as overtones, but instead show an interesting
additive relation, expressed in the Ritz combination principle, and
in addition a still more striking relation involving the squares
of integers, discovered by Balmer. Furthermore, the existence
of stable nonradiating atoms was not to be understood on the
basis of classical theory, for a system consisting of electrons
revolving about atomic nuclei would be expected to emit radiant
energy until the electrons had fallen into the nuclei.
Bohr, no doubt inspired by the work of Einstein mentioned
above, formulated the two following postulates, which to a great
extent retain their validity in the quantum mechanics.
I. The Existence of Stationary States. An atomic system can
exist in certain stationary states f each one corresponding to a
definite value of the energy W of the system; and transition from
one stationary state to another is accompanied by the emission
or absorption as radiant energy, or the transfer to or from
another system, of an amount of energy equal to the difference
in energy of the two states.
II. The Bohr Frequency Rule. The frequency of the radiation
emitted by a system on transition from an initial state of energy
W2 to a final state of lower energy Wi (or absorbed on transition
from the state of energy Wi to that of energy TF 2 ) is given by
the equation 1
1 This relation was suggested by the Ritz combination principle t which it
closely resembles. It was found empirically by Rite and others that if
28 THE OLD QUANTUM THEORY [H5b
W 2 ~ Wi
v =
(51)
Bohr in addition gave a method of determining the quantized
states of motion — the stationary states — of the hydrogen atom.
His method of quantization, involving the restriction of the
angular momentum of circular orbits to integral multiples of
the quantum h/2ir, though leading to satisfactory energy
levels, was soon superseded by a more powerful method, described
in the next section.
Problem 51. Consider an electron moving in a circular orbit about a
nucleus of charge Ze. Show that when the centrifugal force is just balanced
by the centripetal force Ze 2 /r 2 , the total energy is equal to onehalf the
potential energy —Ze 2 /r. Evaluate the energy of the stationary states for
which the angular momentum equals nh/2ir } with n  1, 2, 3, • • • .
6b. The WilsonSommerfeld Rules of Quantization. — In
1915 W. Wilson and A. Sommerfeld discovered independently 1
a powerful method of quantization, which was soon applied,
especially by Sommerfeld and his coworkers, in the discussion
lines of frequencies v\ and vi occur in the spectrum of a given atom it is
frequently possible to find also a line with frequency vi f ^ or v\ — j> 2 .
This led directly to the idea that a set of numbers, called term values, can
be assigned to an atom, such that the frequencies of all the spectral lines
can be expressed as differences of pairs of term values. Term values are
usually given in wave numbers, since this unit, which is the reciprocal
of the wave length expressed in centimeters, is a convenient one for spectro
scopic use. We shall use the symbol v for term values in wave numbers,
reserving the simpler symbol v for frequencies in sec 1 . The normal state
of the ionized atom is usually chosen as the arbitrary zero, and the term
values which represent states of the atom with lower energy than the ion
are given the positive sign, so that the relation between W and v is
~ = 
he
The modern student, to whom the Bohr frequency rule has become common
place, might consider that this rule is clearly evident in the work of Planck
and Einstein. This is not so, however; the confusing identity of the
mechanical frequencies of the harmonic oscillator (the only system discussed)
and the frequency of the radiation absorbed and emitted by this quantized
system delayed recognition of the fact that a fundamental violation of
electromagnetic theory was imperative.
W. Wilson, Phil. Mag. 29, 795 (1915); A. Sommerfeld, Ann. d. Phys.
51, 1 (1916).
H5c] THE ORIGIN OF THE OLD QUANTUM THEORY 29
of the fine structure of the spectra of hydrogen and ionized
helium, their Zeeman and Stark effects, and many other phe
nomena. The first step of their method consists in solving the
classical equations of motion in the Hamiltonian form (Sec. 2),
therefore making use of the coordinates £1, • • • , <?3n and the
canonically conjugate momenta p h • • • , p 3n as the independent
variables. The assumption is then introduced that only those
classical orbits are allowed as stationary states for which the
following conditions are satisfied:
fpkdq k = n k h, k = 1, 2, • • • , 3n; n k = an integer. (52)
These integrals, which are called action integrals, can be calcu
lated only for conditionally periodic systems; that is, for systems
for which coordinates can be found each of which goes through a
cycle as a function of the time, independently of the others.
The definite integral indicated by the symbol f is taken over
one cycle of the motion. Sometimes the coordinates can be
chosen in several different ways, in which case the shapes of the
quantized orbits depend on the choice of coordinate systems, but
the energy values do not.
We shall illustrate the application of this postulate to the
determination of the energy levels of certain specific problems in
Sections 6 and 7.
5c. Selection Rules. The Correspondence Principle. — The
old quantum theory did not provide a satisfactory method of cal
culating the intensities of spectral lines emitted or absorbed by
a system, that is, the probabilities of transition from one sta
tionary state to another with the emission or absorption of a
photon. Qualitative information was provided, however, by an
auxiliary postulate, known as Bohr's correspondence principle,
which correlated the quantumtheory transition probabilities
with the intensity of the light of various frequencies which would
have been radiated by the system according to classical electro
magnetic theory. In particular, if no light of frequency cor
responding to a given transition would have been emitted
classically, it was assumed that the transition would not take
place. The results of such considerations were expressed in
selection rules.
For example, the energy values nhv of a harmonic oscillator
(as given in the following section) are such as apparently to
30 THE OLD QUANTUM THEORY [H6a
permit the emission or absorption of light of frequencies which
are arbitrary multiples (n 2 — ni)p of the fundamental fre
quency vq. But a classical harmonic oscillator would emit only
the fundamental frequency vo f with no overtones, as discussed
in Section 3 ; consequently, in accordance with the correspondence
principle, it was assumed that the selection rule An = ± 1 was
valid, the quantized oscillator being thus restricted to transitions
to the adjacent stationary states.
6. THE QUANTIZATION OF SIMPLE SYSTEMS
6a. The Harmonic Oscillator. Degenerate States. — It was
shown in the previous chapter that for a system consisting of
a particle of mass m bound to the equilibrium position x —
by a restoring force —kx= —4n 2 mv\x and constrained to move
along the x axis the classical motion consists in a harmonic oscilla
tion with frequency v , as described by the equation
x = x sin 2Kvd. (61)
The momentum p x = mx has the value
p z = 2^771^0^0 cos 2wv t, (62)
so that the quantum integral can be evaluated at once:
(bpxdx = f '°m(27rvoa;o cos 2irvot) 2 dt = 2ir 2 v mxl = nh. (63)
The amplitude x is hence restricted to the quantized values
x 0n == {nh/2jr 2 v m}tt. The corresponding energy values are
W n = T +V = 2r 2 m^xJ m (sin 2 2irv t + cos 2 2irv Q t) = 2r 2 mj>lx 2 0n ,
or
W n = nhvo, n = 0, 1, 2,  • • . (64)
Thus we see that the energy levels allowed by the old quantum
theory are integral multiples of hv , as indicated in Figure 61.
The selection rule An = ± 1 permits the emission and absorption
of light of frequency v only.
A particle bound to an equilibrium position in a plane by
restoring forces with different force constants in the x and y
directions, corresponding to the potential function
V = Wm(vlx 2 + v 2 y 2 ) f (fr5)
n6b]
THE QUANTIZATION OF SIMPLE SYSTEMS
31
is similarly found to carry out independent harmonic oscillations
along the two axes. The quantization restricts the energy to
the values
W nx ny = n x hv x + KlyhVy, 71*, Uy = 0, 1, 2,
determined by the two quantum numbers n x and n v
■ , (66)
The ampli
tudes of motion x and y are given by two equations similar to
Equation 63.
/,wi
v,w
n5
V
n4 /
n» 3 /
\ n " 2 /
\ n \ /
V /
A ~^
Fig. 61. — Potentialenergy function and quantized energy levels for the har
monic oscillator according to the old quantum theory.
In case that v x = v v = v , the oscillator is said to be isotropic.
The energy levels are then given by the equation
W n = (n x + riy)hvo = nhv . (67)
Different states of motion, corresponding to different sets of values
of the two quantum numbers n 9 and n v , may then correspond
to the same energy level. Such an energy level is said to be
degenerate, the degree of degeneracy being given by the number
of independent sets of quantum numbers. In this case the nth
level shows (n + l)fold degeneracy. The nth level of the
threedimensional isotropic harmonic oscillator shows
 — fold degeneracy.
6b. The Rigid Rotator. — The configuration of the system of
a rigid rotator restricted to a plane is determined by a single
angular coordinate, say x« The canonically conjugate angular
momentum, p x = Ix, where / is the moment of inertia, 1 is a
1 See Section 36a. footnote, for a definition of moment of inertia.
32 THE OLD QUANTUM THEORY [II6c
constant of the motion. 1 Hence the quantum rule is
f Q *Pxdx = 2rrp x = Kh
or
Vx = g, K = 0, 1, 2, • • • . (68)
Thus the angular momentum is an integral multiple of h/2ir, as
originally assumed by Bohr. The allowed energy values are
v 2 K 2 h 2
W
K=5
The rigid rotator in space can be
described by polar coordinates of
the figure axis, <p and #. On apply
ing the quantum rules it is found
K c 4 that the total angular momentum is
given by Equation 68, and the
component of angular momentum
along the z axis by
P, = ^ M=K,K + l,
K2 • • 7r  ,0, • • • , +K. (610)
K«3
1 ■■ ■ K=i The energy levels are given by
"fig. G~2.~Energy levels iortL Equation 69, each level being
rotator according to the old (2K + l)fold degenerate, inas
quantum theory. much ag the quantum numbe r M
does not affect the energy (Fig. 62).
6c. The Oscillating and Rotating Diatomic Molecule. — A
molecule consisting of two atoms bonded together by forces
which hold them near to the distance r apart may be approxi
mately considered as a harmonic oscillator joined with a rigid
rotator of moment of inertia / = nrf, ju being the reduced mass.
The quantized energy levels are then given by the equation
K 2 h 2
W vK = ^o+J^j> (611)
v being the oscillational or vibrational quantum number 2 and K
1 Section le, footnote.
* The symbol v is now used by band spectroscopists rather than n for this
quantum number.
H6d] THE QUANTIZATION OF SIMPLE SYSTEMS 33
the rotational quantum number. The selection rules for such a
molecule involving two unlike atoms are AK = ±1, Av = ±1.
Actual molecules show larger values of Au, resulting from devi
ation cf the potential function from that corresponding to
harmonic oscillation.
The frequency of light absorbed in a transition from the state
with quantum numbers v" ', K" to that with quantum numbers
t/, K' is
v*>k».,'k' = (v f  v")v +(K'> K">)~,
or, introducing the selection rule AK = ±1,
r,»K».*K»+i = {V  v")"o + (±2K" + 1)~ (612)
The lines corresponding to this equation are shown in Figure 63
for the fundamental oscillational band v = — > v = 1, together
Calculated by equation 6~I2
I I I I I I I I I I I I I I I I I I
K)*? ?*88*7 7*6 6*5.5*44<*3 32 2*1 1*00*1 1*22*3 3*445 5*6 677*8
Observed
10*9 9*8 8*7 7*6 6*5 5*44*3 3*22*1 1*0 0*1 1*22*3 4*5 6*7 8*9
v ^ 3*4 5*6 7*8 9*0
Fig. d3. — The observed rotational fine structure of the hydrogen chloride
fundamental oscillational band v = — > v = 1, showing deviation from the
equidistant spacing of Equation 612.
with th3 experimentally observed absorption band for hydrogen
chloride. It is seen that there is rough agreement; the observed
lines are not equally spaced, however, indicating that our theo
retical treatment, with its assumption of constancy of the moment
of inertia J, is too strongly idealized.
6d. The Particle in a Box. — Let us consider a particle of mass
m in a box in the shape of a rectangular parallelepiped with
edges a, 6, and c, the particle being under the influence of no
34 THE OLD QUANTUM THEORY [H6e
forces except during collision with the walls of the box, from
which it rebounds elastically. The linear momenta p x , p v , and
p t will then be constants of the motion, except that they will
change sign on collision of the particle with the corresponding
walls. Their values are restricted by the rule for quantization
as follows:
(ppxdx = 2ap x = n x h f p x = ~i n x = 0, 1, 2, • • •
P* = 2jr> n, = 0, 1, 2, • • •
Consequently the total energy is restricted to the values
W^. = i(rf + rf + rf)  ^ + = + J*) («W4)
6e. Diffraction by a Crystal Lattice. — Let us consider an
infinite crystal lattice, involving a sequence of identical planes
spaced with the regular interval d. The allowed states of motion
of this crystal along the z axis we assume, in accordance with
the rules of the old quantum theory, to be those for which
jfpgdz = n z h.
For this crystal it is seen that a cycle for the coordinate z is the
identity distance d, so that (p z being constant in the absence of
forces acting on the crystal) the quantum rule becomes
* d n t h
pjlz = nji, or p t = ~ (&15)
I
Any interaction with another system must be such as to leave p 8
quantized; that is, to change it by the amount Ap t = Artgh/d
or nh/d, in which n = An* is an integer. One such type of
interaction is collision with a photon of frequency v> represented
in Figure 64 as impinging at the angle # and being specularly
reflected. Since the momentum of a photon is hv/c, and its
hv
component along the z axis — sin #, the momentum transferred
c
to the crystal is — sin # = ~ sin #. Equating this with the
C A
II6e]
THE QUANTIZATION OF SIMPLE SYSTEMS
35
allowed momentum change of the crystal nh/d, we obtain the
expression
nX = 2d sin #. (616)
This is, however, just the Bragg equation for the diffraction of
xrays by a crystal. This derivation from the corpuscular view
of the nature of light was given
by Duane and Compton 1 in
1923.
Let us now consider a particle,
say an electron, of mass m simi
larly reflected by the crystal.
The momentum transferred to
the crystal will be 2mv sin #,
which is equal to a quantum
for the crystal when
n— = 2d sin *. (617)
mv
64. — The reflection of a photon
by a crystal.
Fia.
Thus we see that a particle would be scattered by a crystal only
when a diffraction equation similar to the Bragg equation for
xrays is satisfied. The wave length of light is replaced by the
expression
h
X =
mv
(618)
which is indeed the de Broglie expression for the wave length
associated with an electron moving with the speed v. This
simple consideration, which might have led to the discovery of
the wave character of material particles in the days when the
old quantum theory had not yet been discarded, was overlooked
at that time.
In the above treatment, which is analogous to the Bragg treat
ment of xray diffraction, the assumption of specular reflection is
made. This can be avoided by a treatment similar to Laue's
derivation of his diffraction equations.
The foregoing considerations provide a simple though perhaps
somewhat extreme illustration of the power of the old quantum
theory as well as of its indefinite character. That a formal argu
ment of this type leading to diffraction equations usually derived
1 W. Duane, Proc. Nat. Acad. Sci. 9, 158 (1923); A. H. Compton, ibid.
9, 359 U923).
36 THE OLD QUANTUM THEORY [H7a
by the discussion of interference and reinforcement of waves
could be carried through from the corpuscular viewpoint with the
old quantum theory, and that a similar treatment could be given
the scattering of electrons by a crystal, with the introduction of
the de Broglie wave length for the electron, indicates that the
gap between the old quantum theory and the new wave mechanics
is not so wide as has been customarily assumed. The indefinite
ness of the old quantum theory arose from its incompleteness —
its inability to deal with any systems except multiplyperiodic
ones. Thus in this diffraction problem we are able to derive
only the simple diffraction equation for an infinite crystal, the
interesting questions of the width of the diffracted beam, the dis
tribution of intensity in different diffraction maxima, the effect
of finite size of the crystal, etc., being left unanswered. 1
7. THE HYDROGEN ATOM
The system composed of a nucleus and one electron, whose
treatment underlies any theoretical discussion of the electronic
structure of atoms and molecules, was the subject of Bohr's first
paper on the quantum theory. 2 In this paper he discussed cir
cular orbits of the planetary electron about a fixed nucleus.
Later 3 he took account of the motion of the nucleus as well as
the electron about their center of mass and showed that with
the consequent introduction of the reduced mass of the two
particles a small numerical deviation from a simple relation
between the spectral frequencies of hydrogen and ionized helium
is satisfactorily explained. Sommerfeld 4 then applied his more
general rules for quantization, leading to quantized elliptical
orbits with definite spatial orientations, and showed that the
relativistic change in mass of the electron causes a splitting of
energy levels correlated with the observed fine structure of
hydrogenlike spectra. In this section we shall reproduce the
Sommerfeld treatment, except for the consideration of the rela
tivistic correction.
7a. Solution of the Equations of Motion. — The system con
sists of two particles, the heavy nucleus, with mass m\ and
1 The application of the correspondence principle to this problem was made
by P. S. Epstein and P. Ehrenfest, Proc. Nat. Acad. Sci. 10, 133 (1924).
* N. Bohr, Phil. Mag. 26, 1 (1913).
8 N. Bohr, ibid. 27, 506 (1914).
4 A. Sommerfeld, Ann. d. Phys. 51, 1 (1916).
H7a] THE HYDROGEN ATOM 37
electric charge +Ze, and the electron, with mass m 2 and charge
— e, between which there is operative an inversesquare attrac
tive force corresponding to the potentialenergy function
V(r) = Ze 2 /r,
r being the distance between the two particles. (The gravi
tational attraction is negligibly small relative to the electro
static attraction.) The system is similar to that of the sun and a
planet, or the earth and moon. It was solved by Sir Isaac
Newton in his " Philosophiae Naturalis Principia Mathematical
wherein he showed that the orbits of one particle relative to the
other are conic sections. Of these we shall discuss only the
closed orbits, elliptical or circular, inasmuch as the old quantum
theory was incapable of dealing with the hyperbolic orbits of the
ionized hydrogen atom.
The system may be described by means of Cartesian coordi
nates X\ 9 2/1, Z\ and x 2 , 2/2, z 2 °f the two particles. As shown in
Section 2d by the introduction of coordinates x, y, z of the center
of mass and of polar coordinates r, #, <p of the electron relative
to the nucleus, the center of mass of the system undergoes
translational motion in a fixed direction with constant speed,
like a single particle in fieldfree space, and the relative motion
of electron and nucleus is that of a particle of mass u = \ >
nti + m a
the reduced mass of the two particles, about a fixed center to
which it is attracted by the same force as that between the
electron, and nucleus. Moreover, the orbit representing any
state of motion lies in a plane (Sec. Id).
In terms of variables r and x in the plane of motion, the
Lagrangian equations of motion are
Ze 2
pr = fxrx 2  p (71)
and
j t (nr*x) = 0. (72)
The second of these can be integrated at once (as in Sec. Id), to
give
ixr 2 x = Vi a constant. (73)
This first result expresses Keoler's area law: The radius vector
38 THE OLD QUANTUM THEORY [H7a
from sun to planet sweeps out equal areas in equal times. The
constant p is the total angular momentum of the system.
Eliminating x horn Equations 71 and 73, we obtain
* r  £3 ~ IT' (74)
which on multiplication by r and integration leads to
The constant of integration W is the total energy of the system
(aside from the translational energy of the system as a whole).
Instead of solving this directly, let us eliminate t to obtain an
equation involving r and x Since
. _ dr _ dr dx _ dr p
~~ dt ~~ dxdt ~~ dx ^
Equation 75 reduces to
(76)
A dr_Y = _1
\r 2 dx) r 2
p'r
or, introducing the new variable
, 2Ze 2 n 2fiW
2 + ^2 r "t — *
u = ; (77)
r
±d X = , d " (78)
p" +"^ W
This can be integrated at once, for W either positive or negative.
In the latter case (closed orbits) there is obtained
1 ZeV , 1 /4 M 2 Z 2 e 4 7 SnW . , . , a ,
«=;=^r+2V~^ + '^ sm(x ~ Xo)  (7 " 9)
This is the equation of an ellipse with the origin at one focus, as
in Figure 71. In terms of the eccentricity e and the semimajor
and semiminor axes a and 6, the equation of such an ellipse is
r a(l  « 2 ) " ^ + 6 2 S n (x Xo) '
(710)
with b = aVl  « 2 .
n7b]
THE HYDROGEN ATOM
39
Thus it is found that the elements of the elliptical orbit are given
by the equations
™ h  P 1  .2 _ J*E£
2W' " v/Z^w ~~ M^V
a = —
(711)
The energy TF is determined by the major axis of the ellipse
alone.
As shown in Problem 51, the total energy for a circular orbit
is equal to onehalf the potential energy and to the kinetic energy
with changed sign. It can be shown also that similar relations
*~*
Fig. 71.
An elliptical electronorbit for the hydrogen atom according to the
old quantum theory.
hold for the timeaverage values of these quantities for elliptic
orbits, that is, that
W = V 2 V = T, (712)
in which the barred symbols indicate the timeaverage values of
the dynamical quantities.
7b. Application of the Quantum Rules. The Energy Levels. —
The WilsonSommerfeld quantum rules, in terms of the polar
coordinates r, #, and <p } are expressed by the three equations
tfprdr = n r h, (713a)
fp*d& = n*h, (7136)
ffydv = mh. (713c)
Since p? is a constant (Sec. le) f the third of these can be integrated
at once, giving
40 THE OLD QUANTUM THEORY P7b
2*p v = mh, or p,g, m=±l, ±2, • • • .
(714)
Hence the component of angular momentum of the orbit along the
z axis can assume only the quantized values which are integral
multiples of h/2ir. The quantum number m is called the mag
netic quantum number, because it serves to distinguish the
various slightly separated levels into which the fieldfree energy
levels are split upon the application of a magnetic field to the
atom. This quantum number is closely connected with the
orientation of the oldquantumtheory orbit in space, a question
discussed in Section Id.
The second integral is easily discussed by the introduction of
the angle x and its conjugate momentum p x = p, the total
angular momentum of the system, by means of the relation,
given in Equation 141, Section le,
Vxdx = P*d& + Pedcp. (715)
In this way we obtain the equation
f Vx d x = kh, (716)
in which p x is a constant of the motion and k is the sum of n&
and m. This integrates at once to
%rp = kh, or p = £"*, k = 1, 2, • • • . (717)
Hence the total angular momentum of the orbit was restricted
by the old quantum theory to values which are integral mul
tiples of the quantum unit of angular momentum h/2w. The
quantum number k is called the azimuthal quantum number.
To evaluate the first integral it is convenient to transform it
in the following way, involving the introduction of the angle %
and the variable u = 1/r with the use of Equation 76:
f^^K$ dx '*•$$**• (7  18)
From Equation 710 we find on differentiation
du _ e cos (x  xo)
d x o(l  « 2 ) '
(719)
II7b] THE HYDROGEN ATOM 41
with the use of which the r quantum condition reduces to the
form
pe 2 p cos2 .(x  xo) d = nX (y^)
Jo {1 + €sm( x  xo)} 2
The definite integral was evaluated by Sommerfeld. 1 The
resultant equation is
^(vf=,  l )  nA (7 " 21)
This, with the value of p of Equation 717 and the relation
b = a\/l — e 2 , leads to the equation
_ _ ___ _ _. ( 7 22 )
In this equation we have introduced a new quantum number n>
called the total quantum number , as the sum of the azimuthal
quantum number k and the radial quantum number n r :
n = n r + k. (723)
With these equations and Equation 711, the energy values
of the quantized orbits and the values of the major and minor
semiaxes can be expressed in terms of the quantum numbers
and the physical constants involved. The energy is seen to
have the value
Wn = **«* . _*«*, (724)
n 2 h 2 n 2 f v 7
being a function of the total quantum number alone. The value
of jffi, the Rydberg constant^ which is given by the equation
R . *£. (725)
depends on the reduced mass \x of the electron and the nucleus.
It is known very accurately, being obtained directly from
spectroscopic data, the values as reported by Birge for hydrogen,
ionized helium, and infinite nuclear mass being
R n = 109,677.759 ± 0.05 cm" 1 ,
R He = 109,722.403 ± 0.05 cm 1 ,
fl w = 109,737.42 ± 0.06 cm" 1 .
1 A. Sommerfeld, Ann. d. Phys. 51, 1 (1916).
42 THE OLD QUANTUM THEORY [H7b
The major and minor semiaxes have the values
n 2 a . nka n .
a = £) o = —=) (726)
««  OT (727)
in which the constant a has the value
The value of this quantity, which for hydrogen is the distance of
the electron from the nucleus in the circular orbit with n = 1,
k = 1, also depends on the reduced mass, but within the experi
mental error in the determination of e the three cases mentioned
above lead to the same value 1
a = 0.52851,
in which 1 A = 1 X 10~ 8 cm. The energy may also be expressed
in terms of a as
Ze 2 Z 2 e 2
The total energy required to remove the electron from the
normal hydrogen atom to infinity is hence
W» = ^ = iW.  £ (729)
This quantity, TT H = 2.1528 X 10"" 11 ergs, is often expressed in
volt electrons, TP H = 13.530 v.e., or in reciprocal centimeters or
wave numbers, Wn = 109,677.76 cm"" 1 (the factor he being
omitted), or in calories per mole, Wn = 311,934 cal/mole.
The energy levels of hydrogen are shown in Figure 72. It is
seen that the first excitation energy, the energy required to raise
the hydrogen atom from the normal state, with n = 1, to the
first excited state, with n = 2, is very large, amounting to
10.15 v.e. or 234,000 cal/mole. The spectral lines emitted by
an excited hydrogen atom as it falls from one stationary state to
another would have wave numbers or reciprocal wave lengths v
given by the equation
p = R (w^ ~ ^)' (7 " 30)
1 The value given by Birge for infinite mass is
0.5281 e* ± 0.0004 X 10" f cm,
that for hydrogen being 0.0003 larger (Appendix I).
H7c]
THE HYDROGEN ATOM
43
in which n" and n' are the values of the total quantum number
for the lower and the upper state, respectively. The series of
lines corresponding to n" = 1, that is, to transitions to the normal
state, is called the Lyman series, and those corresponding to
n" = 2, 3, and 4 are called the Balmer, Paschen, and Brackett
series, respectively. The Lyman series lies in the ultraviolet
region, the lower members of the Balmer series are in the visible
region, and the other series all lie in the infrared.
wo
W«Rhc
n1 ■
Fig. 72.
Lyman
series
—The energy levels of the hydrogen atom, and the transitions giving
rise to the Lyman, Balmer, Paschen, and Brackett series.
7c. Description of the Orbits. — Although the allowed orbits
given by the treatment of Section 76 are not retained in the
quantummechanical model of hydrogen, they nevertheless
serve as a valuable starting point for the study of the more subtle
concepts of the newer theories. The oldquantumtheory orbits
are unsatisfactory chiefly because they restrict the motion too
rigidly, a criticism which is generally applicable to the results of
this theory.
For the simple nonrelativistic model of the hydrogen atom in
fieldfree space the allowed orbits are certain ellipses whose com
mon focus is the center of mass of the nucleus and the electron,
and whose dimensions are certain functions of the quantum
44
THE OLD QUANTUM THEORY
[II7c
numbers, as we have seen. For a given energy level o! the
atom there is in general more than one allowed ellipse, since the
energy depends only on the major axis of the ellipse and not on
its eccentricity or orientation in space. These different ellipses
are distinguished by having different values of the azimuthal
k1
Fig. 73a, b, c
BohrSommerfeld electronorbits for n
to the same scale.
1, 2, and 3, drawn
quantum number fc, which may be any integer from 1 to n.
WTienJ^quidsji^ the orbit is a circle, as is seen from Equation
726. For k less than n, the minor semiaxis b is less than the
major semiaxis a, the eccentricity e of the orbit increasing as k
decreases relative to n. The value zero for k was somewhat
arbitrarily excluded, on the basis of the argument that the
n7d] THE HYDROGEN ATOM 45
corresponding orbit is a degenerate line ellipse which would
cause the electron to strike the nucleus.
Figure 73 shows the orbits for n = 1, 2, and 3 and for the
allowed values of A;. The three different ellipses with n = 3
have major axes of the same length and minor axes which
decrease with decreasing A;. Figure 73 also illustrates the
expansion of the orbits with increasing quantum number, the
radii of the circular orbits increasing as the square of n.
A property of these orbits which is of particular importance in
dealing with heavier atoms is the distance of closest approach of
the electron to the nucleus. Using the expressions for a and b
given in Equation 726 and the properties of the ellipse, we obtain
71 ( 71 —— ■ "\/ 71 —~~ Jc )fLn
for this distance the value — ~ — This formula
and the orbits drawn in Figure 73 show that the most eccentric
orbit for a given n, i.e., that with the smallestvalue. of fc, comes
the nearest to the nucleus. In manyelectron atoms, this
causes a separation of the energies corresponding to these
different elliptical orbits with the same n, since the presence of
the other electrons, especially the inner or core electrons, causes a
modification of the field acting on the electron when it enters
the region near the nucleus.
Since the charge on the nucleus enters the expression for the
radius of the orbit given by Equations 726 and 727, the orbits
for He + are smaller than the corresponding ones for hydrogen,
the major semiaxis being reduced onehalf by the greater charge
on the heliumion nucleus.
7d. Spatial Quantization. — So far we have said nothing of
the orientation of the orbits in space. If a weak field, either
electric or magnetic, is applied to the atom, so that the z direction
in space can be distinguished but no appreciable change in
energy occurs, the z component of the angular momentum of
the atom must be an integral multiple of h/2ir, as mentioned
in Section 76 following Equation 714. This condition, which
restricts the orientation of the plane of the orbit to certain definite
directions, is called spatial quantization. The vector representing
the total angular momentum p is a line perpendicular to the
plane of the orbit (see Sec. le) and from Equation 717 has the
length kh/2r. The z component of the angular momentum is
of length fc cos <a(h/%c) 9 if w is the angle between the vector p and
46 THE OLD QUANTUM THEORY [H7d
the z axis. This results in the following expression for cos o>:
m
cos co = r •
The value zero for m was excluded for reasons related to those
used in barring k = 0, so that m may be ±1, ±2, • • • , ±k.
m+2
(m»0)
m»3
Fig. 74a, b, c. — Spatial quantization of BohrSommerfeld orbits with A; = 1, 2.
and 3.
For the lowest state of hydrogen, in which k = 1 (and for all orbits
for which k = 1), there are only two values of m, +1 and — 1,
which correspond to motion in the xy plane in a counterclockwise
or in a clockwise sense. For fc = 2 four orientations are per
D8] THE DECLINE OF THE OLD QUANTUM THEORY 47
mitted, as shown in Figure 74. Values ±k for m always cor
respond to orbits lying in the xy plane.
It can be shown by the methods of classical electromagnetic
theory that the motion of an electron with charge — e and mass
kh
m in an orbit with angular momentum ^ gives rise to a magnetic
kh 6
field corresponding to a magnetic dipole of magnitude ^^ —
oriented in the same direction as the angular momentum vector.
The component of magnetic moment in the direction of the z axis
he
is m A The energy of magnetic interaction of the atom with
47rra c
he
a magnetic field of strength H parallel to the z axis is mj — H.
It was this interaction energy which was considered to give rise
to the Zeeman effect (the splitting of spectral lines by a magnetic
field) and the phenomenon of paramagnetism. It is now known
that this explanation is only partially satisfactory, inasmuch as
the magnetic moment associated with the spin of the electron,
discussed in Chapter VIII, also makes an important contribution.
he
The magnetic moment j — I s called a Bohr magneton.
Problem 71. Calculate the frequencies and wave lengths of the first
five members of the Balmer series for the isotopic hydrogen atom whose
mass is approximately 2.0136 on the atomic weight scale, and compare with
those for ordinary hydrogen.
Problem 72. Quantize the system consisting of two neutral particles
of masses equal to those of the electron and proton held together by gravita
tional attraction, obtaining expressions for the axes of the orbits and the
energy levels.
8. THE DECLINE OF THE OLD QUANTUM THEORY
The historical development of atomic and molecular mechanics
up to the present may be summarized by the following division
into periods (which, of course, are not so sharply demarcated as
indicated) :
19131920. The origin and extensive application of the old
quantum theory of the atom.
19201925. The decline of the old quantum theory.
1925 . The origin of the new quantum mechanics and
its application to physical problems.
48 THE OLD QUANTUM THEORY [H8
1927 The application of the new quantum mechanics
to chemical problems.
The present time may well be also the first part of the era of the
development of a more fundamental quantum mechanics, includ
ing the theory of relativity and of the electromagnetic field, and
dealing with the mechanics of the atomic nucleus as well as of the
extranuclear structure.
vThe decline of the old quantum theory began with the introduc
tion of halfintegral values for quantum numbers in place of
integral values for certain systems, in order to obtain agreement
with experiment* It was discovered that the pure rotation
spectra of the hydrogen halide molecules are not in accordance
with Equation 69 with K = 0, 1, 2, • • • , but instead require
K = }i, %, • • • . Similarly, half integral values of the oscilla
tional quantum number v in Equation 611 were found to be
required in order to account for the observed isotope displace
ments for diatomic molecules. Halfintegral values for the
azimuthal quantum number k were also indicated by observations
on both polarization and penetration of the atom core by a
valence electron. Still more serious were cases in which agree
ment with the observed energy levels could not be obtained by
the methods of the old quantum theory by any such subterfuge
or arbitrary procedure (such as the normal state of the helium
atom, excited states of the helium atom, the normal state of
the hydrogen molecule ion, etc.), and cases where the methods
of the old quantum theory led to definite qualitative disagreement
with experiment (the influence of a magnetic field on the dielectric
constant of a gas, etc.). Moreover, the failure of the old quan
tum theory to provide a method of calculating transition probabil
ities and the intensities of spectral lines was recognized more
and more clearly as a fundamental flaw. Closely related to this
was the lack of a treatment of the phenomenon of the disper
sion of light, a problem which attracted a great amount of
attention.
This dissatisfaction with the old quantum theory culminated
in the formulation by Heisenberg 1 in 1925 of his quantum
mechanics, as a method of treatment of atomic systems leading
to values of the intensities as well as frequencies of spectral
lines. The quantum mechanics of Heisenberg was rapidly
1 W. Heisenberg, Z. /. Phys. 33, 879 (1925).
H8] THE DECLINE OF THE OLD QUANTUM THEORY 49
developed by Heisenberg, Born, and Jordan 1 by the introduction
of matrix methods. In the meantime Schrodinger had inde
pendently discovered and developed his wave mechanics, 2
stimulated by the earlier attribution of a wave character to the
electron by de Broglie 3 in 1924. The mathematical identity of
matrix mechanics and wave mechanics was then shown by
Schrodinger 4 and by Eckart. 5 The further development of the
quantum mechanics was rapid, especially because of the con
tributions of Dirac, who formulated 6 a relativistic theory of the
electron and contributed to the generalization of the quantum
mechanics (Chap. XV).
General References on the Old Quantum Theory
A. Sommerfeld: " Atomic Structure and Spectral Lines," E. P. Dutton &
Co., Inc., New York, 1923.
A. E. Ruark and H: C. Urey: "Atoms, Molecules and Quanta," McGraw
Hill Book Company, Inc., New York, 1930.
1 M. Born and P. Jordan, ibid. 34, 858 (1925); M. Born, W. Heisenberg,
and P. Jordan, ibid. 35, 557 (1926).
2 E. Schrodinger, Ann. d. Phys. 79, 361, 489; 80, 437; 81, 109 (1926).
3 L. de Broglie, Thesis, Paris, 1924; Ann. de phys. (10) 3, 22 (1925).
4 E. Schrodinger, Ann. d. Phys. 79, 734 (1926).
6 C. Eckart, Phys. Rev. 28, 711 (1926).
•P. A. M. Dirac, Proc. Roy. Soc. A 113, 621; 114, 243 (1927); 117, 610
(1928).
CHAPTER III
THE SCHRODINGER WAVE EQUATION WITH THE
HARMONIC OSCILLATOR AS AN EXAMPLE
In the preceding chapters we have given a brief discussion of
the development of the theory of mechanics before the discovery
of the quantum mechanics. Now we begin the study of the quan
tum mechanics itself, starting in this chapter with the Schrodinger
wave equation for a system with only one degree of freedom, the
general principles of the theory being illustrated by the special
example of the harmonic oscillator, which is treated in great
detail because of its importance in many physical problems.
The theory will then be generalized in the succeeding chapter
to systems of point particles in threedimensional space.
9. THE SCHR6DINGER WAVE EQUATION
In the first paragraph of his paper 1 Quantisierung als Eigen
wertproblem, communicated to the Annalen der Physik on
January 27, 1926, Erwin Schrodinger stated essentially:
In this communication I wish to show, first for the simplest case of
the nonrelativistic and unperturbed hydrogen atom, that the usual
rules of quantization can be replaced by another postulate, in which
there occurs no mention of whole numbers. Instead, the introduction
of integers arises in the same natural way as, for example, in a vibrating
string, for which the number of nodes is integral. The new conception
can be generalized, and I believe that it penetrates deeply into the true
nature of the quantum rules.
In this and four other papers, published during the first half of
1926, Schrodinger communicated his wave equation and applied
it to a number of problems, including the hydrogen atom, the
harmonic oscillator, the rigid rotator, the diatomic molecule, and
1 E. SchrOdingbr, Ann. d. Phys. 79, 361 (1926), and later papers referred
to on the preceding page. An English translation of these papers has
appeared under the title E. Schrodinger, "Collected Papers on Wave
Mechanics," Blackie and Son, London and Glasgow, 1928.
60
ra9] the schrOdinger WAVE EQUATION 51
the hydrogen atom in an electric field (Stark effect). For the
last problem he developed his perturbation theory, and for
the discussion of dispersion he also developed the theory of a
perturbation varying with the time. His methods were rapidly
adopted by other investigators, and applied with such success
that there is hardly a field of physics or chemistry that has
remained untouched by Schrodinger's work.
Schrodinger's system of dynamics differs from that of Newton,
Lagrange, and Hamilton in its aim as well as its method. Instead
of attempting to find equations, such as Newton's equations,
which enable a prediction to be made of the exact positions and
velocities of the particles of a system in a given state of motion,
he devised a method of calculating a function of the coordinates
of the system and the time (and not the momenta or velocities),
with the aid of which, in accordance with the interpretation
developed by Born, 1 probable values of the coordinates and
of other dynamical quantities can be predicted for the system.
It was later recognized that the acceptance of dynamical equa
tions of this type involves the renunciation of the hope of describ
ing in exact detail the behavior of a system. The degree of
accuracy with which the behavior of a system can be discussed
by quantummechanical methods forms the subject of Heisen
berg y s uncertainty 'principle? to which we shall recur in Chapter
XV.
The Schrodinger wave equation and its auxiliary postulates
enable us to determine certain functions ^ of the coordinates of a
system and the time. These functions are called the Schrodinger
wave junctions or probability amplitude functions. The square
of the absolute value of a given wave function is interpreted as
a probability distribution function for the coordinates of the
system in the state represented by this wave function, as will
be discussed in Section 10a. The wave equation has been
given this name because it is a differential equation of the second
order in the coordinates of the system, somewhat similar to the
wave equation of classical theory. The similarity is not close,
however, and we shall not utilize the analogy in our exposition.
Besides yielding the probability amplitude or wave function ty f
the Schrodinger equation provides a method of calculating values
1 M. Born, Z. f. Phys. 37, 863; 38, 803 (1926).
* W. Heisenberg, Z. /. Phys. 43, 172 (1927).
52 THE SCHRODINGER WAVE EQUATION [HI9
of the energy of the stationary states of a system, the existence
of which we have discussed in connection with the old quantum
theory. No arbitrary postulates concerning quantum numbers
are required in this calculation; instead, integers enter auto
matically in the process of finding satisfactory solutions of the
wave equation.
For our purposes, the Schrodinger equation, the auxiliary
restrictions upon the wave function % and the interpretation of
the wave function are conveniently taken as fundamental
postulates, with no derivation from other principles necessary.
This idea may be clarified by a comparison with other branches
of physics. Every department of deductive science must
necessarily be founded on certain postulates which are regarded
as fundamental. Frequently these fundamental postulates are
so closely related to experiment that their acceptance follows
directly upon the acceptance of the experiments upon which
they are based, as, for example, the inversesquare law of electrical
attraction. In other cases the primary postulates are not so
directly obvious from experiment, but owe their acceptance to the
fact that conclusions drawn from them, often by long chains of
reasoning, agree with experiment in all of the tests which have
been made. The second law of thermodynamics is representative
of this type of postulate. It is not customary to attempt to
derive the second law for general systems from anything more
fundamental, nor is it obvious that it follows directly from
some simple experiment; nevertheless, it is accepted as correct
because deductions made from it agree with experiment. It is
an assumption, justified only by the success achieved by its
consequences.
The wave equation of Schrodinger belongs to this latter class
of primary assumption. It is not derived from other physical
laws nor obtained as a necessary consequence of any experiment;
instead, it is assumed to be correct, and then results predicted
by it are compared with data from the laboratory.
A clear distinction must frequently be made between the way
in which a discoverer arrives at a given hypothesis and the
logical position which this hypothesis occupies in the theory when
it has been completed and made orderly and deductive. In
the process of discovery, analogy often plays a very important
part. Thus the analogies between geometrical optics and
m9a] THE SCHRODINGER WAVE EQUATION 53
classical mechanics on the one hand and undulatory optics and
wave mechanics on the other may have assisted Schrodinger to
formulate his now famous equation; but these analogies by no
means provide a logical derivation of the equation.
In many cases there is more than one way of stating the funda
mental postulates. Thus either Lagrange's or Hamilton's form
of the equations of motion may be regarded as fundamental for
classical mechanics, and if one is so chosen, the other can be
derived from it. Similarly, there are other ways of expressing
the basic assumptions of quantum mechanics, and if they are
used, the wave equation can be derived from them, but, no
matter which mode of presenting the theory is adopted, some
starting point must be chosen, consisting of a set of assumptions
not deduced from any deeper principles.
It often happens that principles which have served as the basis
for whole branches of theory are superseded by other principles
of wider applicability. Newton's laws of motion, adopted
because they were successful in predicting the motions of the
planets and in correlating celestial and terrestrial phenomena,
were replaced by Lagrange's and Hamilton's equations because
these are more general. They include Newton's laws as a
special case and in addition serve for the treatment of motions
involving electric, magnetic, and relativistic phenomena. Like
wise, quantum mechanics includes Newton's laws for the special
case of heavy bodies and in addition is successful in problems
involving atoms and electrons. A still more general theory
than that of Schrodinger has been developed (we shall discuss
it in Chap. XV), but for nearly all purposes the wave equation is a
convenient and sufficient starting point.
9a. The Wave Equation Including the Time. — Let us first
consider a Newtonian system with one degree of freedom,
consisting of a particle of mass m restricted to motion along a
fixed straight line, which we take as the x axis, and let us assume
that the system is further described by a potentialenergy func
tion V (x) throughout the region — <x> < x < + «> . For this
system the Schrodinger wave equation is assumed to be
8r 2 m dx 2 ' w v ' ' 2iri dt
In this equation the function ^(z, t) is called the Schrddinger
54 THE SCHRODINGER WAVE EQUATION [m9a
wave function including the time, or the probability amplitude
function. It will be noticed that the equation is somewhat
similar in form to the wave equations occurring in other branches
of theoretical physics, as in the discussion of the motion of a
vibrating string. The student facile in mathematical physics
may well profit from investigating this similarity and also the
analogy between classical mechanics and geometrical optics on
the one hand, and wave mechanics and undulatory optics on the
other. 1 However, it is not necessary to do this. An extensive
previous knowledge of partial differential equations and their
usual applications in mathematical physics is not a necessary
prerequisite for the study of wave mechanics, and indeed the
study of wave mechanics may provide a satisfactory introduction
to the subject for the more physically minded or chemically
minded student.
The Schrodinger time equation is closely related to the equation
of classical Newtonian mechanics
H(p„ x) = T(p.) + V(z) = W, (92)
which states that the total energy W is equal to the sum of the
kinetic energy T and the potential energy V and hence to the
Hamiltonian function H(p x , x). Introducing the coordinate x
and momentum p x , this equation becomes
ff(Px, x) = JLp» + V(x) = W. (93)
If we now arbitrarily replace p x by the differential operator
h A h A
s— . — and W by — sr zv and introduce the function ¥(x, t) on
2m dx Van dt
which these operators can operate, this equation becomes
H (hh x )
* ( *> ()  "ass w + v *  33 aT (9_4)
which is identical with Equation 91. The wave equation is
1 See, for example, Condon and Morse, "Quantum Mechanics," p.
10, McGrawHill Book Company, Inc., New York, 1929; Ruark and Urey,
"Atoms, Molecules and Quanta," Chap. XV, McGrawHill Book Company,
Inc., New York, 1930; E. Schr6dinger, Ann. d. Phys. 79, 489 (1926); K. K.
Darrow, Rev. Mod. Phys. 6, 23 (1934); or other treatises on wave mechanics,
listed at the end of this chapter.
m9a] THE SCHRODINOER WAVE EQUATION 55
consequently often conveniently written as
H* « W*, (95)
h A l a
in which it is understood that the operators s— r and — tt— ^
are to be introduced.
Iji replacing p x by the operator — ; — > p* is to be replaced by ( — . ) — >
* 2m dx \2m/ dx*
and so on. (In some cases, which, however, do not arise in the simpler
problems which we are discussing in this book, there may be ambiguity
regarding the formulation of the operator. 1 ) It might be desirable to dis
tinguish between the classical Hamiltonian function H = H{p Xt x) and the
Hamiltonian operator
H
\2m dx /
as by writing /^operator for the latter. We shall not do this, however, since
the danger of confusion is small. Whenever H is followed by ^ (or by ^,
representing the wave functions not including the time, discussed in the
following sections), it is understood to be the Hamiltonian operator. Simi
h d
larly, whenever W is followed by ^ it represents the operator : —
2t% dt
The symbol W will also be used to represent the energy constant (Sees.
96, 9c). We shall, indeed, usually restrict the symbol W to this use, and
h * , .
wnte — for the operator.
2iri dt
It must be recognized that this correlation of the wave equation
and the classical energy equation, as well as the utilization
which we shall subsequently make of many other classical
dynamical expressions, has only formal significance. It provides
a convenient way of describing the system for which we are
setting up a wave equation by making use of the terminology
developed over a long period of years by the workers in classical
dynamics. Thus our store of direct knowledge regarding the
nature of the system known as the hydrogen atom consists in the
results of a large number of experiments — spectroscopic, chemical,
etc. It is found that all of the known facts about this system
can be correlated and systematized (and, we say, explained)
by associating with this system a certain wave equation. Our
confidence in the significance of this association increases when
predictions regarding previously uninvestigated properties of
1 B. Podolsky, Phys. Rev. 32, 812 (1928).
56 THE SCHR6DINGER WAVE EQUATION [III9b
the hydrogen atom are subsequently verified by experiment.
We might then describe the hydrogen atom by giving its wave
equation; this description would be complete. It is unsatis
factory, however, because it is unwieldy. On observing that
there is a formal relation between this wave equation and the
classical energy equation for a system of two particles of different
masses and electrical charges, we seize on this as providing a
simple, easy, and familiar way of describing the system, and
we say that the hydrogen atom consists of two particles, the
electron and proton, which attract each other according to
Coulomb's inversesquare law. Actually we do not know that
the electron and proton attract each other in the same way
that two macroscopic electrically charged bodies do, inasmuch
as the force between the two particles in a hydrogen atom has
never been directly measured. All that we do know is that the
wave equation for the hydrogen atom bears a certain formal
relation to the classical dynamical equations for a system of
two particles attracting each other in this way.
Having emphasized the formal nature of this correlation and
of the usual description of wavemechanical systems in terms of
classical concepts, let us now point out the extreme practical
importance of this procedure. It is found that satisfactory wave
equations can be formulated for nearly all atomic and molecular
systems by accepting the descriptions of them developed during
the days of the classical and old quantum theory and translating
them into quantummechanical language by the methods
discussed above. Indeed, in many cases the wavemechanical
expressions for values of experimentally observable properties of
systems are identical with those given by the old quantum theory,
and in other cases only small changes are necessary. Throughout
the following chapters we shall make use of such locutions as
"a system of two particles with inversesquare attraction"
instead of "a system whose wave equation involves six coordi
nates and a function e 2 / r i2," etc.
9b. The Amplitude Equation. — In order to solve Equation 91,
let us (as is usual in the solution of a partial differential equation
of this type) first study the solutions ^ (if any exist) which can
be expressed as the product of two functions, one involving the
time alone and the other the coordinate alone :
¥(*, t) * HxMt).
m9b] THE SCHRODINGER WAVE EQUATION 57
On introducing this in Equation 91 and dividing through by
4s(x)<p(t), it becomes
The right side of this equation is a function of the time t alone
and the left side a function of the coordinate x alone. It is
consequently necessary that the value of the quantity to which
each side is equal be dependent on neither x nor t; that is, that
it be a constant. Let us call it W. Equation 96 can then
be written as two equations, namely,
_ = —wxo
and \ (97)
The second of these is customarily written in the form
2 2 + 8 p{W 7(*))*0, (98)
obtained on multiplying by — 87r 2 ra//i 2 and transposing the term
in IF.
Equation 98 is often itself called the Schrodinger wave equa
tion, or sometimes the amplitude equation, inasmuch as \l/(x)
determines the amplitude of the function >£(#, t). It is found
that the equation possesses various satisfactory solutions, cor
responding to various values of the constant W. Let us indicate
these values of W by attaching the subscript n, and similarly
represent the amplitude function corresponding to W n as \p n (x).
The corresponding equation for <p(t) can be integrated at once
to give
w
The general solution of Equation 91 is the sum of all the particu
lar solutions with arbitrary coefficients. We consequently
write as the general expression for the wave function for this
system
*(*, t) = %an*n(x, t)  2a n * n (s)e~ 2 ""A (910)
w^ (99)
58 THE SCHRODINGER WAVE EQUATION [m9c
in which the quantities a n are constants. The symbol V is
n
to be considered as representing the process of summation over
discrete values of W n or integration over a continuous range or
both, according to the requirements of the particular case.
It will be shown later that the general postulates which we
shall make regarding the physical interpretation of the wave
function require that the constant W n represent the energy of
the system in its various stationary states.
9c. Wave Functions. Discrete and Continuous Sets of
Characteristic Energy Values. — The functions ^W which
satisfy Equation 98 and also certain auxiliary conditions, dis
cussed below, are variously called wave functions or eigenf unctions
(Eigenfunktionen), or sometimes amplitude functions, charac
teristic functions, or proper functions. It is found that satis
factory solutions \l/ n of the wave equation exist only for certain
values of the parameter W n (which is interpreted as the energy
of the system). These values W n are characteristic energy values
or eigenvalues (Eigenwerte) of the wave equation. A wave
equation of this type is called a characteristic value equation.
Inasmuch as we are going to interpret the square of the absolute
value of a wave function as having the physical significance of a
probability distribution function, it is not unreasonable that the
wave function be required to possess certain properties, such as
singlevaluedness, necessary in order that this interpretation be
possible and unambiguous. It has been found that a satisfactory
wave mechanics can be constructed on the basis of the following
auxiliary postulates regarding the nature of wave functions:
To be a satisfactory wave function, a solution of the Schrodinger
wave equation must be continuous, singlevalued, and finite 1 through
1 The assumption that the wave function be finite at all points in configura
tion space may be more rigorous than necessary. Several alternative
postulates have been suggested by various investigators. Perhaps the most
satisfying of these is due to W. Pauli ("Handbuch der Physik," 2d ed., Vol.
XXVI, Part 1, p. 123). In Section 10 we shall interpret the function ^*^ as a
probability distribution function. In order that this interpretation may be
made, it is necessary that the integral of ^*SIr over configuration space be a
constant with changing time. Pauli has shown that this condition is satis
fied provided that ¥ is finite throughout configuration space, but that it is
also satisfied in certain cases by functions which are not finite everywhere.
The exceptional cases are rare and do not occur in the problems treated iD
this book.
m9c]
THE SCHRODINGER WAVE EQUATION
59
out the configuration space of the system (that is, for all values of
the coordinate x which the system can assume).
These conditions are those usually applied in mathematical
physics to functions representing physical quantities. For
example, the function representing the displacement of a vibrat
ing string from its equilibrium configuration would have to
satisfy them.
For a given system the characteristic energy values W n may
occur only as a set of discrete values, or as a set of values covering
a continuous range, or as both. From analogy with spectroscopy
it is often said that in these three cases the energy values comprise
a discrete spectrum, a continuous spectrum, or both. The way
w 3
\v Wj
X
W,
w x>v
• W ° V
•xa
Fia. 91. — Potentialenergy function for a general system with one degree of
freedom.
in which the above postulates regarding the wave equation and
its acceptable solutions lead to the selection of definite energy
values may be understood by the qualitative consideration of a
simple example. Let us consider, for our system of one degree
of freedom, that the potentialenergy function V(x) has the form
given in Figure 91, such that for very large positive or negative
values of x, V(x) increases without limit. For a given value of
the energy parameter W, the wave equation is
dH
*8vm
W}+.
(911)
dx 2 h 2
In the region of large x (x > a) the quantity V(x)  W will be
positive. Hence in this region the curvature ^ wil1 be positive
if $ is positive, and negative if \p is negative. Now let us assume
60
THE SCHRODINQER WAVE EQUATION
[in9c
that at an arbitrary point x = c the function \j/ has a certain value
(which may be chosen arbitrarily, inasmuch as the wave equation
is a homogeneous equation 1 ) and a certain slope p> as indicated
for Curve 1 in Figure 92. The behavior of the function, as it
is continued both to the right and to the left, is completely
determined by the values assigned to two quantities; to wit, the
slope j at the point x = c, and the energy parameter W in the
wave equation, which determines the value of the second deriva
Fig. 92. — The behavior of \p for x > a.
tive. As we have drawn Curve 1, the curvature is determined
by the wave equation to be negative in the region x < a, where
V(x) — W is negative, \p being positive, and hence the curve can
be continued to the right as shown. At the point x = a, the
function remaining positive, the curvature becomes positive, the
curve then being concave upward. If the slope becomes positive,
as indicated, then the curve will increase without limit for
increasing x, and as a result of this "infinity catastrophe'' the
function will not be an acceptable wave function.
1 An equation is homogeneous in \j/, if the same power of \J/ (in our case the
first power) occurs in every term. The function obtained by multiplying
any solution of a homogeneous equation by a constant is also a solution
of the equation.
III9c] THE SCHRODINGER WAVE EQUATION 61
We can now make a second attempt, choosing the slope at
x = c as indicated for Curve 3. In this case the curve as drawn
intersects the x axis at a point x = d to the right of a. For
values of x larger than d the function \p is negative, and the curva
ture is negative. The function decreases in value more and more
rapidly with increasing x, again suffering the infinity catastrophe,
and hence it too is not an acceptable wave function in this
region.
Thus we see that, for a given value of W, only by a very careful
selection of the slope of the function at the point x = c can the
function be made to behave properly for large values of x. This
selection, indicated by Curve 2, is such as to cause the wave
function to approach the value zero asymptotically with increas
ing x.
Supposing that we have in this way determined, for a given
value of W y a value of the slope at x = c which causes the
function to behave properly for large positive values of x, we
extend the function to the left and consider its behavior for large
negative values of x. In view of our experience on the right,
it will not be surprising if our curve on extension to the left
behaves as Curve 1 or Curve 3 on the right, eliminating the
function from consideration; in fact, it is this behavior which
is expected for an arbitrarily chosen value of W. We can now
select another value of W for trial, and determine for it the value
of the slope at x = c necessary to cause the function to behave
properly on the right, and then see if, for it, the curve behaves
properly on the left also. Finally, by a very careful choice of
the value of the energy parameter W, we are able to choose a
slope at x = c which causes the function to behave properly
both for very large and for very small values of x. This value
of W is one of the characteristic values of the energy of the
system. In view of the sensitiveness of the curve to the param
eter W, an infinitesimal change from this satisfactory value will
cause the function to behave improperly.
We conclude that the parameter W and the slope at the point
x = c (for a given value of the function itself at this point) can
have only certain values if \p is to be an acceptable wave function.
For each satisfactory value of W there is one (or, in certain
cases discussed later, more than one) satisfactory value of the
slope, by the use of which the corresponding wave function can
62
THE SCHRODINGER WAVE EQUATION
[in9c
be built up. For this system the characteristic values W n
of the energy form a discrete set, and only a discrete set, inasmuch
as for every value of W, no matter how large, V{x) — W is
positive for sufficiently large positive or negative values of x.
It is customary to number the characteristic energy values for
such a system as indicated in Figure 91, W being the lowest,
W\ the next, and so on, corresponding to the wave functions
y l / o(x) t ^i(z), etc. The integer n, which is written as a subscript
in W n and ^„(a:), is called the quantum number. For such a
onedimensional system it is equal to the number of zeros 1
possessed by \l/ n . A slight extension of the argument given above
W
orJi
V'
X>
Fia. 93. — The energy levels for a system with V( — <») or F(}oo) finite.
shows that all of the zeros lie in the region between the points
x = b and x = a, outside of which V(x) — W n remains positive.
The natural and simple way in which integral quantum numbers
are introduced and in which the energy is restricted to definite
values contrasts sharply with the arbitrary and uncertain
procedure of the old quantum theory.
Let us now consider a system in which the potentialenergy
function remains finite at x •—► + °° or at x — > — oo or at both
limits, as shown in Figure 93. For a value of W smaller than
both F(+<») and F(—oo) the argument presented above is
valid. Consequently the energy levels will form a discrete set
for this region. If W is greater than F (+«>), however, a
similar argument shows that the curvature will be such as always
to return the wave function to the x axis, about which it will
1 A zero of ^*(x) is a point {x = xi) at which \J/ n is equal to zero.
mlOa] PHYSICAL INTERPRETATION OF WAVE FUNCTIONS 63
oscillate. Hence any value of W greater than F(+°°) or
V{ — oo ) will be an allowed value, corresponding to an acceptable
wave function, and the system will have a continuous spectrum
of energy values in this region.
9d. The Complex Conjugate Wave Function W*(x, t).—ln
the physical interpretation of the wave equation and its solutions,
as discussed in the following section, the quantity ^*(x, t),
the complex conjugate of ^(x, t), enters on an equivalent basis
with V(x, t). The wave equation satisfied by V* is the complex
conjugate of Equation 91, namely,
_ h* a*9*(x, t) v , (x w (x t) _± a**(*, (Q _ l2)
8^i dx* +V W * {X ' l)  2ri dt (9 12)
The general solution of this conjugate wave equation is the
following, the conjugate of 910:
*•(*, = £<*:*„*(*, = XaMix)**^. (913)
n n
(Some authors have adopted the convention of representing
by the symbol Mr the wave function which is the solution of
Equation 912 and by ^* that of 91. This is only a matter of
nomenclature.)
It will be noticed that in the complex conjugate wave function
the exponential terms containing the time are necessarily different
from the corresponding terms in ^ itself, the minus sign being
removed to form the complex conjugate. The amplitude
functions $n(x) } on the other hand, are frequently real, in which
case ^*(x) = fn(x).
10. THE PHYSICAL INTERPRETATION OF THE WAVE FUNCTIONS
10a. W*(x, t)V(x, t) as a Probability Distribution Function —
Let us consider a given general solution ^(x, t) of the wave equa
tion. For a given value of the time t, the function V*(x, t)¥(x, t) y
the product of ¥ and its complex conjugate, is a function defined
for all values of x between — oo and + » ; that is, throughout the
configuration space of this onedimensional system. We now
make the following postulate regarding the physical significance
of *:
The quantity V*(x, t)V(x, t)dx is the probability thai the system
in the physical situation represented by the wave function V(x t t)
64 THE SCHRODINOER WAVE EQUATION lIHlOb
have at the time t the configuration represented by a point in the
region dx of configuration space. In other words, ty*(x, t)^(x, t)
is a probability distribution junction for the configuration of the
system. In the simple system under discussion, ^*(x, t)V(x, t)dx
is the probability that the particle lie in the region between x
and x + dx at the time t.
In order that this postulate may be made, the wave function
^{x y t) must be normalized to unity (or, briefly, normalized);
that is, the constants a n of Equation 910 must be so chosen as
to satisfy the relation
* + "Vfo t)*(x, t)dx = 1, (101)
/
inasmuch as the probability that the coordinate x of the particle
lie somewhere between — oo and +oo is necessarily unity.
It is also convenient to normalize the individual amplitude
functions ^ n (x) to unity, so that each satisfies the equation
* + °Vn* (x)Mx)dx = 1. (102)
J
Moreover, as proved in Appendix III, it is found that the
independent solutions of any amplitude equation can always be
chosen in such a way that for any two of them, \l/ m (x) and yp n {x) }
the integral $$m(x)yl/ n (x)dx over all of configuration space van
ishes; that is,
f*y%(x)+ n (x)dx =0, m * n. (103)
The functions are then said to be mutually orthogonal. Using
these relations and Equations 910 and 913, it is found that a
wave function ty(x, t) = Va n ^ n (a;, t) is normalized when the
n
coefficients a n satisfy the relation
X a nX = 1. (104)
n
10b. Stationary States. — Let us consider the probability dis
tribution function ty*ty for a system in the state represented by
the wave function ^(x y t) = ^a n \l/ n (x)e ** h and its conjugate
n
S£*(x, t) = 2^a*\l/*(x)e h On multiplying these series
nilOc] PHYSICAL INTERPRETATION OF WAVE FUNCTIONS 65
together, ^*^ is seen to have the form
M (W.W.) f
A
in which the prime on the doublesummation symbol indicates
that only terms with m ?± n are included. In general, then,
the probability function and hence the properties of the system
depend on the time, inasmuch as the time enters in the exponen
tial factors of the double sum. Only if the coefficients a n are
zero for all except one value of W n is ^*^ independent of t.
In such a case the wave function will contain only a single term
(with n = n'j say) Sfv(x, t) = i/ n '(x)e h , the amplitude
function i/v(z) being a particular solution of the amplitude
equation. For such a state the properties of the system as given
by the probability function ^*^ are independent of the time, and
the state is called a stationary state.
10c. Further Physical Interpretation. Average Values of
Dynamical Quantities. — If we inquire as to what average value
would be expected on measurement at a given time t of the
coordinate x of the system in a physical situation represented by
the wave function ^, the above interpretation of ^*^ leads to the
answer
•f 00
y*(x, t^ixy t)xdx\
f
that is, the value of x is averaged over all configurations, using
the function SF*^ as a weight or probability function. A similar
integral gives the average value predicted for x 2 , or x 3 , or any
function F(x) of the coordinate x:
p= r*"JV(3, t)*(x, t)F(x)dx. (105)
In order that the same question can be answered for a more
general dynamical function G(p x , x) involving the momentum p x
as well as the coordinate x, we now make the following more
general postulate:
The average value of the dynamical function G(p Xf x) predicted
for a system in the physical situation represented by the wave
function V(x, t) is given by the integral
66
THE SCHRODINGER WAVE EQUATION
[m10c
= f + "**(*, Og/^. ±, * W «)d«, (106)
in which the operator (?, obtained from ©(p*, a:) by replacing p x
h /J
by 5—. — > operates on the function ^f{x } t) and the integration is
extended throughout the configuration space of the system. 1
In general, the result of a measurement of G will not be given
by this expression for G. G rather is the average of a very
large number of measurements made on a large number of
identical systems in the physical situation represented by ^, or
repeated on the same system, which before each measurement
must be in the same physical situation. For example, if SF is
Fig. 101. — Two types of probability distribution function ^e*9,
finite for a range of values of x (Curve A } Figure 101), then a
measurement of x might lead to any value within this range,
the probability being given by SF*^. Only if ^*^ were zero
for all values of x except x = a, as indicated by Curve B in
Figure 101, would the probability of obtaining a particular
value x = a on measurement of x be unity. In this case the
value a r would be predicted with probability unity to be obtained
on measurement of the rth power of x; so that for such a prob
ability distribution function x r is equal to (x) r . It has also
been shown by mathematicians that the existence of this identity
of G r and (G) r for all values of r is sufficient to establish that the
probability distribution function for the dynamical quantity G is
of type B) that is, that the value of G can be predicted accurately.
1 In some cases further considerations are necessary in order to determine
the exact form of the operator, but we shall not encounter such difficulties.
mlla] HARMONIC OSCILLATOR IN WAVE MECHANICS 67
Even if the system is in a stationary state, represented by the
wave function V n (x, t) = \l/ n (x)e h , only an average value
can be predicted for an arbitrary dynamical quantity. The
energy of the system, corresponding to the Hamiltonian function
H(Px, x), has, however, a definite value for a stationary state
of the system, equal to the characteristic value W n found on
solution of the wave equation, so that the result of a measurement
of the energy of the system in a given stationary state can be
predicted accurately. To prove this, we evaluate H r and {B) r .
H is given by the integral
the factor involving the time being equal to unity. This trans
forms with the use of Equation 98 into
^ = /J 1 " y:(x)Wn+n(x)dx,
\l/*(x)\l/ n (x)dx = 1,
H = W n , and (H) r  W n . (107)
By a similar procedure, involving repeated use of Equation 98,
it is seen that W is equal to W r n . We have thus shown H r to
be equal to (H) r , in consequence of which, in accordance with the
argument set forth above, the energy of the system has the
definite value W n .
Further discussion of the physical significance of wave functions
will be given in connection with the treatment of the harmonic
oscillator in this chapter and of other systems in succeeding
chapters, and especially in Chapter XV, in which the question
of deciding which wave function to associate with a given system
under given circumstances will be treated. In the earlier sections
we shall restrict the discussion mainly to the properties of
stationary states.
11. THE HARMONIC OSCILLATOR IN WAVE MECHANICS
11a. Solution of the Wave Equation. — As our first example
of the solution of the Schrodinger wave equation for a dynamical
system we choose the onedimensional harmonic oscillator, not
only because this provides a good illustration of the methods
68 THE SCHRODINGER WAVE EQUATION [Hllla
employed in applying the wave equation, but also because this
system is of considerable importance in applications which we
shall discuss later, such as the calculation of the vibrational
energies of molecules. The more difficult problem of the three
dimensional oscillator was treated by the methods of classical
mechanics in Section la, while the simple onedimensional
case was discussed according to the old quantum theory in
Section 6a.
The potential energy may be written, as before, in the form
V(x) = 2w 2 mv$x 2 , in which x is the displacement of the particle
of mass m from its equilibrium position x = 0. Insertion of this
in the general wave equation for a onedimensional system
(Eq. 98) gives the equation
g + ^(W  arWoZ*)* = 0, (111)
or, introducing for convenience the quantities X = 8w 2 mW/h 2
and a = 47r 2 m^ //i,
g + (X  «»**)* = 0. (112)
We desire functions yp{x) which satisfy this equation throughout
the region of values — <x> to + °° for x ) and which are acceptable
wave functions, i.e., functions which are continuous, single
valued, and finite throughout the region. A straightforward
method of solution which suggests itself is the use of a power
series expansion for ^, the coefficients of the successive powers
of x being determined by substitution of the series for yp in the
wave equation. There is, however, a very useful procedure
which we may make use of in this and succeeding problems,
consisting of the determination of the form of ^ in the regions of
large positive or negative values of x } and the subsequent dis
cussion, by the introduction of a factor in the form of a power
series (which later reduces to a polynomial), of the behavior of ^
for \x\ small. This procedure may be called the polynomial
method. 1
The first step is the asymptotic solution of the wave equation
when  a;  is very large. For any value of the energy constant W,
a value of x can be found such that for it and all larger values
1 A. Sommbbfeld, "Wave Mechanics," p. 11.
mlla] HARMONIC OSCILLATOR IN WAVE MECHANICS 69
of \x\, X is negligibly small relative to <xV, the asymptotic form
of the wave equation thus becoming
2* = «V*. (113)
This equation is satisfied asymptotically by the exponential
functions
\b = P. 2
inasmuch as the derivatives of \p have the values
= ±<xxe
#_ ^..^i*
dx
and
•Ts = a 2 x 2 e 2 + ae 2
ax 2 ~
and the second term in ^— 2 is negligible in the region considered.
Of the two asymptotic solutions e 2 and e 2 } the second is
unsatisfactory as a wave function since it tends rapidly to
infinity with increasing values of \x\; the first, however, leads to a
satisfactory treatment of the problem.
We now proceed to obtain an accurate solution of the wave
equation throughout configuration space (—«)<£<+«>),
based upon the asymptotic solution, by introducing as a factor
a power series in x and determining its coefficients by substitution
in the wave equation.
Let $ = e~* xt f(x). Then
= e~* Xi {a*x>f a} 2axf +/"},
in which /' and /" represent p and r~> respectively. Equation
1 12 then becomes, on division by e 2 ,
f>  2axf + (X  a)} = 0, (114)
the terms in a 2 x 2 f cancelling.
70 THE SCHRODINGER WAVE EQUATION [mlla
It is now convenient to introduce a new variable { , related to
x by the equation
i = V**, (115)
and to replace the' function f(x) by H(£), to which it is equal.
The differential equation 114 then becomes
2 *f + C 1 ) H =  ^
dm
dp
We now represent H (£) as a power series, which we differentiate
to obtain its derivatives,
v
^ = 2" ( " " 1)a ^'" 2 = 1 • 2 «2 + 2 • 3a 3 £ + • • • .
v
On substitution of these expressions, Equation 116 assumes the
following form:
1 • 2a 2 + 2 • 3a 3 £ + 3 • 4a 4 £ 2 + 4 • 5a 5 £ 3 + ■ • ■
 2ai£  2 • 2a 2 £ 2  2 • 3a 3 £ 3  • • •
+0 i ) + (s i )'" f+ C i )'" £!+
a>)
«,{» + • • • =0.
In order for this series to vanish for all values of £ (i.e., for #(£)
to be a solution of 116), the coefficients of individual powers of
£ must vanish separately 1 :
1 • 2oi + Q  1 Ja„ = 0,
2 • 3a 3 + Q  1  2 V = 0,
3 • 4a 4 + Q  1  2 • 2 Ja 2 = 0,
4 • 5a 6 + Q  1  2 • 3 Ja 8 = 0,
1 See footnote, Sec. 23.
mlla] HARMONIC OSCILLATOR IN WAVE MECHANICS 71
or, in general, for the coefficient of $",
(v + !)(* + 2K +2 + Q  1  2vja y =
or
(=»)
a ' +2 = („ + !)(„ + 2)°" (1I_7)
This expression is called a recursion formula. It enables the
coefficients a 2 , a 8 , a 4 , • • • to be calculated successively in
terms of a and a\, which are arbitrary. If a is set equal to zero,
only odd powers appear; with a± zero, the series contains even
powers only.
For arbitrary values of the energy parameter X, the above
given series consists of an infinite number of terms and does
not correspond to a satisfactory wave function, because, as we
shall show, the value of the series increases too rapidly as x
increases, with the result that the total function, even though it
includes the negative exponential factor, increases without
limit as x increases. To prove this we compare the series for
H and that for e ft ,
£4 £6 tv tv+2
eP = 1 + I 2 + h + h + • • • +tV + 7 s — n + • • • ■
2! ' 3!
©' (H<
For large values of J the first terms of these series will be unim
portant. Suppose that the ratio of the coefficients of the vth
terms in the expansion of H (J) and e? is called c, which may be
small or large, i.e., a y /b y = c, if b y is the coefficient of £" in the
expansion of e*\ For large enough values of v, we have the
asymptotic relations
so that
a v4 .2 = —a>v and b v +i = —b P9
v v
CLy+2 0t V
by+2 by
if v is large enough. Therefore, the higher terms of the series for
H differ from those for e** only by a multiplicative constant, so
that for large values of £, for which the lower terms are unim
72 THE SCHRODINGER WAVE EQUATION [IIIlla
portant, H will behave like e? and the product e 2 H will behave
like e 2 in this region, thus making it unacceptable as a wave
function.
We must therefore choose the values of the energy parameter
which will cause the series for H to break off after a finite number
of terms, leaving a polynomial. This yields a satisfactory wave
_I 2
function, because the negative exponential factor e 2 will cause
the function to approach zero for large values of £. The value
of X which causes the series to break off after the nth term is
seen from Equation 117 to be
X = (2n + l)a. (118)
It is, moreover, also necessary that the value either of a or of a i
be put equal to zero, according as n is odd or even, inasmuch as a
suitably chosen value of X can cause either the even or the odd
series to break off, but not both. The solutions are thus either
odd or even functions of £. This condition is a sufficient condi
tion to insure that the wave equation 112 have satisfactory
solutions, and it is furthermore a necessary condition; no other
values of X lead to satisfactory solutions. For each integral
value 0, 1, 2, 3, • • • of n, which we may call the quantum
number of the corresponding state of the oscillator, a satisfactory
solution of the wave equation will exist. The straightforward
way in which the quantum number enters in the treatment of
the wave equation, as the degree of the polynomial #(£), is
especially satisfying when compared with the arbitrary assump
tion of integral or halfintegral multiples of h for the phase
integral of the old quantum theory.
The condition expressed in Equation 118 for the existence
of the nth wave function becomes
W = W n = (n + H)hv , n = 0, 1, 2, • • • , (119)
when X and a are replaced by the quantities they represent. A
comparison with the result W = nhv obtained in Section 6a
by the old quantum theory shows that the only difference is
that all the energy levels are shifted upward, as shown in Figure
11— 1, by an amount equal to half the separation of the energy
levels, the socalled zeropoint energy %hv . From this we
Hilib] HARMONIC OSCILLATOR IN WAVE MECHANICS 73
see that even in its lowest state the system has an energy greater
than that which it would have if it were at rest in its equilibrium
position. The existence of a zeropoint energy, which leads to
an improved agreement with experiment, is an important feature
of the quantum mechanics and recurs in many problems. 1
Just as in the oldquantumtheory treatment, the frequency
emitted or absorbed by a transition between adjacent energy
levels is equal to the classical vibration frequency v (Sec. 40c).
V
n5
n4 /
i I
n3 /
VJY
n2 /
n1 /
\ n«0 /
V y
Fig. 111. — Energy levels for the harmonic oscillator according to wave me*
chanics (see Fig. 61).
lib. The Wave Functions for the Harmonic Oscillator and
Their Physical Interpretation. — For each of the characteristic
values W n of the energy, a satisfactory solution of the wave
equation 111 can be constructed by the use of the recursion
formula 117. Energy levels such as these, to each of which
there corresponds only one independent wave function, are said
to be nondegenerate to distinguish them from degenerate energy
levels (examples of which we shall consider later), to which several
1 The name zeropoint energy is used for the energy of a system in its lowest
stationary state because the system in thermodynamic equilibrium with its
environment at a temperature approaching the absolute zero would be in
this stationary state. The zeropoint energy is of considerable importance
in many statisticalmechanical and thermodynamic discussions. The
existence of zeropoint energy is correlated with the uncertainty principle
(Chap. XV),
74
THE SCHRODINOER WAVE EQUATION
mllb]
independent wave functions correspond. The solutions of 111
may be written in the form
*.(*) =N n e~W n (S), (1110)
in which £ = \/ax. H n (£) is a polynomial of the nth degree in
£ , and N n is a constant which is adjusted so that \p n is normalized,
i.e., so that 4/ n satisfies the relation
f*y:(x)M*)dx = 1,
(11H)
in which \f/* 9 the complex conjugate of ^ n , is in this case equal to
^ n . In the next section we shall discuss the nature and properties
Vol&i
V
\
« ■
j i
> 
(
)
\
[*M\
_zr
1 I
55
3u
CX
v ^t
^7 ~E_
1
1
3 2
«* **
Fig. 112. — The wave function ^o(£) for the normal state of the harmonic
oscillator (left), and the corresponding probability distribution function
[^o(£)P (right). The classical distribution function for an oscillator with the
same total energy is shown by the dashed curve.
of these solutions \p n in great detail. The first of them, which
corresponds to the state of lowest energy for the system, is
**(*) = ($ e ~ 2 = (irf e ~**' (11 " 12)
Figure 112 shows this function. From the postulate discussed
in Section 10a, ^J^ = ^§, which is also plotted in Figure 112,
represents the probability distribution function for the coordinate
x. In other words, the quantity \[/l(x)dx at any point x gives
the probability of finding the particle in the range dx at that
point. We see from the figure that the result of quantum
mechanics for this case does not agree at all with the probability
function which is computed classically for a harmonic oscillator
with the same energy. Classically the particle is most likely to
mllbj HARMONIC OSCILLATOR IN WAVE MECHANICS 75
be found at the ends of its motion, which are clearly defined points
(the classical probability distribution is shown by the dotted
curve in Figure 112), whereas \j/l has its maximum at the origin
of a: and, furthermore, shows a rapidly decreasing but nevertheless
finite probability of finding the particle outside the region allowed
classically. This surprising result, that it is possible for a
particle to penetrate into a region in which its total energy is less
than its potential energy, is closely connected with Heisenberg's
nd
/\
A
£j
 n.,f ^
^\ i "
V J
\t
Xtr
432101234
i
(\
/N
L
1
\
/
1
\
/
t
H
^
\
\
\
\
\
t.
"^
4

! ',
I '
c
)
i :
5 i
\
A
/
\
A
/
/'
J
y
i
\
\
(
/
1
\
/
u
\
/
V
£
\ ",
? ;
> i
r*i,„.
1 '
I s
\ 4
i
*a$
 1
n«<
»
*~
■A
5 
2 
(
)
?
3 ^
I
^
4t
R B
nn^
±s
=*#
£==
it
4 3 21
2 3 4
432101234
Fig. 113. — The wave functions ^ n (£), n — 1 to 6, for the harmonic oscillator.
For each case the heavy horizontal line indicates the region traversed by the
classical harmonic oscillator with the same total energy.
uncertainty principle, which leads to the conclusion that it is
not possible to measure exactly both the position and the velocity
of a particle at the same time. We shall discuss this phenomenon
further in Chapter XV. It may be mentioned at this point,
however, that the extension of the probability distribution func
tion into the region of negative kinetic energy will not require
that the law of the conservation of energy be abandoned.
The form of \p n for larger values of n is shown in Figure 113.
Since H n is a polynomial of degree n, \f/ n will, have n zeros or
points where \p n crosses the zero line. The probability of finding
the particle at these points is zero. Insoection of Figure 113
shows that all the solutions plotted show a general behavior in
76
THE SCHRODINGER WAVE EQUATION
[mllb
agreement with that obtained by the general arguments of Section
9c; that is, inside the classically permitted region of motion of
the particle (in which V{x) is less than W n ) the wave function
oscillates, having n zeros, while outside that region the wave
function falls rapidly to zero in an exponential manner and has
no zeros. Furthermore, we see in this example an illustration
of still another general principle: The larger the value of n, the
Fio. 114. — The probability distribution function [^io(£)] 2 for the state
n ■■ 10 of the harmonic oscillator. Note how closely the function approximates
in its average value the probability distribution function for the classical har
monic oscillator with the same total energy, represented by the dashed curve.
more nearly does the wavemechanical probability distribution
function approximate to the classical expression for a particle
with the same energy. Figure 114 shows \l/ 2 (x) for the state
with n = 10 compared with the classical probability curve for
21
the harmonic oscillator with the same value yhv for the energy.
It is seen that, aside from the rapid fluctuation of the wave
mechanical curve, the general agreement of the two functions
is good. This agreement permits us to visualize the motion of
the particle in a wavemechanical harmonic oscillator as being
mllc] HARMONIC OSCILLATOR IN WAVE MECHANICS 77
similar to its classical toandfro motion, the particle speeding
up in the center of its orbit and slowing down as it approaches
its maximum displacement from its equilibrium position. The
amplitude of the oscillation cannot be considered to be constant,
as for the classical oscillator ; instead, we may picture the particle
as oscillating sometimes with very large amplitude, and some
times with very small amplitude, but usually with an amplitude
in the neighborhood of the classical value for the same energy.
Other properties of the oscillator also are compatible with this
picture; thus the wavemechanical rootmeansquare value of the
momentum is equal to the classical value (Prob. 114).
A picture of this type, while useful in developing an intuitive
feeling for the wavemechanical equations, must not be taken
too seriously, for it is not completely satisfactory. Thus it
cannot be reconciled with the existence of zeros in the wave
functions for the stationary states, corresponding to points where
the probability distribution function becomes vanishingly
small.
lie. Mathematical Properties of the Harmonic Oscillator
Wave Functions. — The polynomials H n (£) and the functions
_v
e 2 H n (£) obtained in the solution of the wave equation for the
harmonic oscillator did not originate with Schrodinger's work
but were well known to mathematicians in connection with other
problems. Their properties have been intensively studied.
For the present purpose, instead of developing the theory of
the polynomials H n (£), called the Hermite polynomials, from the
relation between successive coefficients given in Equation 117,
it is more convenient to introduce them by means of another
definition:
We shall show later that this leads to the same functions as
Equation 117. A third definition involves the use of a generating
function, a method which is useful in many calculations and which
is also applicable to other functions. The generating function
for the Hermite polynomials is
CO
S(«, 8) m e« <«>' m 2^T 8B  (U14)
f8>0
78 THE SCHR6DINQER WAVE EQUATION [Hllic
This identity in the auxiliary variable s means that the function
e*t (•{>* hag foe property that, if it is expanded in a power series
in s, the coefficients of successive powers of s are just the Hermite
polynomials H n (£), multiplied by 1/n !. To show the equivalence
of the two definitions 1113 and 1114, we differentiate S n times
with respect to s and then let s tend to zero, using first one and
then the other expression for S; the terms with v < n vanish
on differentiation, and those with v > n vanish for s — ■> 0, leaving
only the term with v = n:
\ r /»>0
and
Comparing these two equations, we see that we obtain Equation
1113, so that the two definitions of H n (t) are equivalent.
Equation 1113 is useful for obtaining the individual functions,
while Equation 1114 is frequently convenient for deriving their
properties, such as in the case we shall now discuss.
To show that the functions we have defined above are the
same as those used in the solution of the harmonic oscillator
problem, we look for the differential equation satisfied by
fln(£). It is first convenient to derive certain relations between
successive Hermite polynomials and their derivatives. We
note that since S = e pi»v* 9 its partial derivative with respect
to s is given by the equation
g  *.  «*
Similarly differentiating the series S = ^.—^ps", and equating
the two different expressions for dS/ds f we obtain the equation
»■(«).,.
_ 2< 8  »^e.
^J(n  1)!° " y ° "^ n!
n n
or, collecting terms corresponding to the same power of s,
mllc] HARMONIC OSCILLATOR IN WAVE MECHANICS 79
^J ( n! (n — 1)1 n! J
n
Since this equation is true for all values of s, the coefficients of
individual powers of s must vanish, giving as the recursion
formula for the Hermite polynomials the expression
JW0  2(H n (S) + 2n/7 n _ 1 (£)  0. (1115)
Similarly, by differentiation with respect to {, we derive the
equation
f  2sS,
which gives, in just the same manner as above, the equation
n
or
».'(*) = ^P = 2ntf n __ 1 (£), (1116)
involving the first derivatives of the Hermite polynomials.
This can be further differentiated with respect to £ to obtain
expressions involving higher derivatives.
Equations 1115 and 1116 lead to the differential equation for
H n (0, for from 1116 we obtain
HM) = 2nHU(Q = 4n(n  l)ffi(0, (1117)
while Equation 1115 may be rewritten as
# n (£)  2£# n _ 1 ({) + 2(n  l)ff n  2 (£) = 0, (1118)
which becomes, with the use of Equations 1116 and 1117,
or
Hi'(«)  2«Hi(© + 2nH n (S) = 0. (1119)
This is just the equation, 116, which we obtained from the har
monic oscillator problem, if we put 2n in place of 1, as
required by Equation 118. Since for each integral value of n
this equation has only one solution with the proper behavior at
80 THE SCHRODINQER WAVE EQUATION [mllc
infinity, the polynomials H n (g) introduced in Section 11a are
the Hermite polynomials.
The functions
M*) = Nne 2 • H n (& 9 $ = Vi, (1120)
are called the Hermite orthogonal functions; they are, as we
have seen, the wave functions for the harmonic oscillator. The
ypl{x)dx = 1, i.e., which normalizes
The functions are mutually orthogonal if the integral over
configuration space of the product of any two of them vanishes:
f*yn(x)+m(x)dx = 0, n*m. (1122)
To prove the orthogonality of the functions and to evaluate the
normalization constant given in Equation 1121, it is convenient
to consider two generating functions:
and
s & s) = 2^r 8 " = ei ""~ v '
m, t) = 2%r' m = eP ~ lt ~ v>
Using these, we obtain the relations
n m
= f g0iHtoHVtVdt = e 2 " f er«— »•<*({  st)
Considering coefficients of 8 n t m in the two equal series expansions,
HniOHmifie^dZ vanishes for m^n y and has the
value 2 n n!\/ir for m = n, in consequence of which the functions
mllc] HARMONIC OSCILLATOR IN WAVE MECHANICS 81
are orthogonal and the normalization constant has the value
given above.
The first few Hermite polynomials are
Ho(0 =1
ffitt) = 4? 2
h*(q = 8?  m
H A (Q = 16€ 4  48? + 12
H b (0 = 32?  160? + 120f (1123)
ff s (Q = 64?  480? + 720?  120
JJ 7 (€) = 128?  1344? + 3360?  1680?
H n (Q = 256?  3584? + 13440?  13440? + 1680
H 9 (& = 512?  9216? + 48384?  80640? + 30240?
Hio(& = 1024?°  23040? + 161280?  403200? + 302400?
 30240.
The list may easily be extended by the use of the recursion
formula, Equation 1115. Figure 113 shows curves for the
first few wave functions, i.e., the functions given by Equation
1120.
By using the generating functions £ and T we can evaluate
certain integrals involving \[/ n which are of importance. For
example, we may study the integral which, as we shall later
show (Sec. 40c), determines the probability of transition from the
state n to the state m. This is
x nm = tnKxdx = ^^ H n H m e*m. (1124)
J— w OL J— ao
Using S and T we obtain the relation
n m
/»+ 00 /»+ 00
(*  s  t)d(S 8t)
e ( «'°y(? st).
The first integral vanishes, and the second gives Vx On
expanding the exponential, we obtain
82 THE SCHRODINOER WAVE EQUATION [Hilie
V^(
}
Hence, comparing coefficients of s n t m , we see that x nm is zero
except for m = n ± 1, its values then being
Cn.« + i = ^^T (ll25a)
and
It will be shown later that this result requires that transitions
occur only between adjacent energy levels of the harmonic
oscillator, in agreement with the conclusion drawn from the
correspondence principle in Section 5c.
Problem 111. Show that if V( — x) = V(x), with V real, the solutions
yp n {x) of the amplitude equation {M* have the property that ^»( —x) — ± y// n (x) .
Problem 112. Evaluate the integrals
(X*)nm = J>„^m£ 2 dx, (x*) nn = ^n^mX i dx 1 (x*)nm = fMrnZ^dx,
where ^ ft is a solution of the wave equation for the harmonic oscillator.
Problem 113. Calculate the average values of x, x 2 , x*, and x* for a
harmonic oscillator in the nth stationary state. Is it true that £* = (£)*
or that P = (P) 2 ? What conclusions can be drawn from these results
concerning the results of a measurement of x?
Problem 114. Calculate the average values of p x and p* for a harmonic
oscillator in the nth stationary state and compare with the classical values
for the same total energy. From the results of this and of the last problem,
compute the average value of the energy W = T f V for the nth
stationary state.
Problem 115. a. Calculate the zerppoint energy of a system consisting
of a mass of 1 g. connected to a fixed point by a spring which is stretched
1 cm. by a force of 10,000 dynes. The particle is constrained to move only
in the x direction.
b. Calculate the quantum number of the system when its energy is about
equal to kT, where k is Boltzmann's constant and T = 298° A. This corre
sponds to thermodynamic equilibrium at room temperature (Sec. 49).
General References on Quantum Mechanics
A. SoMMBRTBfcD: "Wave Mechanics," E. P. Dutton <fe Co., Inc., New
York, 1930.
E. XJ. Condon and P. M. Morse: "Quantum Mechanics," McGrawHill
Book Company, Inc., New York, 1929.
IIIllc] HARMONIC OSCILLATOR IN WAVE MECHANICS 83
A. E. Ruark and H. C. Urey: "Atoms, Molecules and Quanta," McGraw
Hill Book Company, Inc., New York, 1930.
N. F. Mott: "An Outline of Wave Mechanics," Cambridge University
Press, Cambridge, 1930.
J. Frenkbl: "Wave Mechanics," Oxford University Press, 19331934.
K. K. D arrow: Elementary Notions of Quantum Mechanics, Rev. Mod.
Phys. 6, 23 (1934).
E. C. Kemblb: General Principles of Quantum Mechanics, Part I, Rev.
Mod. Phys. 1, 157 (1929).
E. C. Kemble: "Fundamental Principles of Quantum Mechanics," Mc
GrawHill Book Company, Inc., 1937.
E. C. Kemble and E. L. Hill: General Principles of Quantum Mechanics,
Part II, Rev. Mod. Phys. 2, 1 (1930).
S. Dushman: "Elements of Quantum Mechanics," John Wiley & Sons,
Inc., J 938.
CHAPTER IV
THE WAVE EQUATION FOR A SYSTEM OF POINT
PARTICLES IN THREE DIMENSIONS
12. THE WAVE EQUATION FOR A SYSTEM OF POINT PARTICLES
The Schrodinger equation for a system of N interacting point
particles in threedimensional space is closely similar to that for
the simple onedimensional system treated in the preceding
chapter. The time equation is a partial differential equation
in 3iV + 1 independent variables (the 3N Cartesian coordinates,
say, of the N particles, and the time) instead of only two inde
pendent variables, and the wave function is a function of these
3N + 1 variables. The same substitution as that used for the
simpler system leads to the separation of the time equation into
an equation involving the time alone and an amplitude equation
involving the ZN coordinates. The equation involving the time
alone is found to be the same as for the simpler system, so that
the time dependency of the wave functions for the stationary
states of a general system of point particles is the same as for
the onedimensional system. The amplitude equation, however,
instead of being a total differential equation in one independent
variable, is a partial differential equation in 3N independent
variables, the 3A r coordinates. It is convenient to say that this
is an equation in a 3iVdimensional configuration space, meaning
by this that solutions are to be found for all values of the 3N
Cartesian coordinates X\ • • • z N from — oo to + °° • The
amplitude function, dependent on these ZN coordinates, is said
to be a function in configuration space. A point in configuration
space corresponds to a definite value of each of the 3N coordi
nates Xi  • • z N , and hence to definite positions of the N particles
in ordinary space, that is, to a definite configuration of the
system.
The wave equation, the auxiliary conditions imposed on the
wave functions, and the physical interpretation of the wave
functions for the general system are closely similar to those for
84
IV12a] WAVE EQUATION FOR A SYSTEM OF PARTICLES 85
the onedimensional system, the only changes being those conse
quent to the increase in the number of dimensions of configura
tion space. A detailed account of the postulates made regarding
the wave equation and its solutions for a general system of point
^articles is given in the following sections, together with a dis
cussion of various simple systems for illustration.
12a. The Wave Equation Including the Time. — Let us con
sider a system consisting of N point particles of masses mi,
rri2i • • • , ™>n moving in threedimensional space under the
influence of forces expressed by the potential function V(xi
2/i • • • Zn, t), xi • • • z N being the 3N Cartesian coordinates
of the N particles. The potential function V, representing
the interaction of the particles with one another or with an
external field or both, may be a function of the 3iV coordinates
alone or may depend on the time also. The former case, with
V — V(xi  • • z N ), corresponds to a conservative system.
Our main interest lies in systems of this type, and we shall soon
restrict our discussion to them.
We assume with Schrodinger that the wave equation for this
system is
N
/i 2 %n l/d 2 * , d 2 * , a 2 *\ , TrT h d* /10 ,,
SH^im^ + M + *T ) + F * = "53 Hi' (12 " 1}
This equation is often written as
N
It'
in which v, ? is the Laplace operator or Laplacian for the ith
particle. 1 In Cartesian coordinates, it is given by the expression
Vl ~ dxf ^ dy? ^ dz\
The wave function ^ = V(xi • • • z*, t) is a function of the
3JV coordinates of the system and the time.
It will be noted that the Schrodinger time equation for this
general system is formally related to the classical energy equation
in the same way as for the onedimensional system of the preced
^he symbol A is sometimes used in place of V 2 . The symbol V 2 is
commonly read as del squared.
86 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV12b
ing chapter. The energy equation for a Newtonian system of
point' particles is
H(Vx l • ' ' Pz N , xi • • • z N) t) = T(p Xi • • • p, N ) +
V(xi • • • z N) t) = TF, (122)
which on explicit introduction of the momenta p xi . . . p*^
becomes
ff(p. t • • • p v * • • ■ ** = 2 2 i ( ^ + ^ + ^ +
7(si ■ • • zn, t) = F. (123)
We now arbitrarily replace the momenta p^ • • • p ZN by
the differential operators —.—... _  — , respectively,
2in dxi 2ki dz N
h rk
and W by the operator — jr— . — , and introduce the function
2m dt
V(xi • • • z N , t) on which these operators can operate. The
equation then becomes
„/ h d h d \
H \2ritel ' ' ' 2ridT N ,Xl ' ' '"''J*
= __^^! V 2* + w = _ A «* (124)
which is identical with Equation 121. Just as for the one
dimensional case, the wave equation is often symbolically
written
HV = IF*. (125)
The discussion in Section 9a of the significance of this formal
relation is also appropriate to this more general case.
12b. The Amplitude Equation. — Let us now restrict our atten
tion to conservative systems, for which V is a function of the 3iV
coordinates only. To solve the wave equation for this case,
we proceed exactly as in the simpler problem of Section 96,
investigating the solutions * of the wave equation which can be
expressed as the product of two functions, one of which involves
only the time and the other only the 3N coordinates:
*(xi •••**,*)= *(xi • • • zs)<p(t). (126)
On introducing this expression in Equation 121, the wave equa
IV12b] WAVE EQUATION FOR A SYSTEM OF PARTICLES 87
tion can be separated into two equations, one for <p(t) and one
for }p(xi • • • Zn). These equations are
at n
"^ K ) (127)
£2^ + v + = F ^
i1
The second of these is often written in the form
N
2^ + 8 £ {W  m = ° (1 *~ 8)
*i
This is Schrodinger's amplitude equation for a conservative
system of point particles.
The auxiliary conditions which must be satisfied by a solution
of the amplitude equation in order that it be an acceptable wave
function are given in Section 9c. These conditions must hold
throughout configuration space, that is, for all values between
— oo and + oo for each of the ZN Cartesian coordinates of the
system. Just as for the onedimensional case, it is found that
acceptable solutions exist only for certain values of the energy
parameter W. These values may form a discrete set, a con
tinuous set, or both.
It is usually found convenient to represent the various succes
sive values of the energy parameter and the corresponding ampli
tude functions by the use of SN integers, which represent 3N
quantum numbers rti • • • n ZN , associated with the 3N coordi
nates. The way in which this association occurs will be made
clear in the detailed discussion of examples in the following
sections of this chapter and in later chapters. For the present
let us represent all of the quantum numbers n\ • • ■ n ZN by
the one letter n, and write instead of W ni • * • % w an< i iki " * ' »w
the simpler symbols W n and \fr n . x^ K >
The equation for <p(t) gives on integration L ^
.Wn
2wiZrt
_„.£< _ V J (129)
<p(t) = e * , v
exactly as for the onedimensional system. The various particu
lar solutions of the wave equation are hence
88 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV12d
*n(xi • • • z N , t) = + n (xi ' • • z N )e h . (1210)
These represent the various stationary states of the system. The
general solution of the wave equation is
¥(xi • • • Zn> t) = ^a n ^ n (xi • • • z N , t) =
n
2lrtrf
2/a»(aa • • • z^)e , (1211)
n
in which the quantities a n are constants. The symbol V repre
n
sents summation for all discrete values of W n and integration over
all continuous ranges of values.
12c. The Complex Conjugate Wave Function W*(xi • • • zn>
0 — The complex conjugate wave function V*(xi • • • zx, t)
is a solution of the conjugate wave equation
N
£2£***<* •*«*> +
t1
ssl**** •••«*•'>■ (12 " 12)
The general solution of this equation for a conservative system is
**(Zi • • • Zn, t) = ^*^*(Xi • • • Sat, =
n
2a.v:(*. ■ • • m« * • (12 " 13)
12d. The Physical Interpretation of the Wave Functions. —
The physical interpretation of the wave functions for this general
system is closely analogous to that for the onedimensional system
discussed in Section 10. We first make the following postulate,
generalizing that of Section 10a:
The quantity ^*(xi • • • z N , t)^(xi • • • z N , t)dx\ • • • dz N is
the probability that the system in the physical situation represented
by the wave function ty(x i • • • Zn, t) have at the timet the configura
tion represented by a point in the volume element dx\ • • • dzN of
configuration space. ¥ *¥ thus serves as a probability distribution
function for the configuration of the system.
IV12d] WAVE EQUATION FOR A SYSTEM OF PARTICLES 89
The function V(xi • • • z N , t) must then be normalized to
unity, satisfying the equation
S**(zi • • • z Ny 0*(*i • • z N , t)dr = 1, (1214)
in which the symbol dr is used to represent the volume element
dxi • • • dz N in configuration space, and the integral is to* be
taken over the whole of configuration space. (In the remaining
sections of this book the simple integral sign followed by dr
is to be considered as indicating an integral over the whole of
configuration space.) It is also convenient to normalize the
amplitude functions \p n (xi • • • z N ), according to the equation
JVn*(*i • • • z»)M*i • • • *»)d* = 1. (1215)
It is found, as shown in Appendix III, that the independent
solutions of any amplitude equation (just as for the onedimen
sional case) can be chosen in such a way that any two of them are
orthogonal, satisfying the orthogonality equation
jVtffri • • • z N )+ n (*i ' ' • z N )dr = 0, m^n. (1216)
A wave function ty(xi • • • z N > t) = ^\a n ^ n {xi • • • zs, t) is
n
then normalized if the coefficients a n satisfy the equation
2<*X = 1. (1217)
n
An argument analogous to that of Section 10b shows that the
wave functions V n (xi • • • z N , t) = \p n (xi • • • z N )e h give
probability distribution functions which are independent of the
time and hence correspond to stationary states.
A more general physical interpretation can be given the wave
functions, along the lines indicated in Section 10c, by making
the postulate that the average value of the dynamical function
G(p Xl • • • p tJf) xi • • • z N , t) predicted for a system in the
physical situation represented by the wave function V(xi • • • z N ,
t) is given by the integral
in which the operator G, obtained from G(p Xl
h
2iri
d
dz N y
Xi • •
• z Ny t)
• •
' **:
t 0*>
(1218)
[P*x
. . .
V*N> Zl
• • • Zjfj
90 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV13
h A h A
t) by replacing p„ . . . p,„by_— . . . ——respectively,
operates on the function ^(xi • • • z Ny t) and the integration is
extended throughout the configuration space of the system.
Further discussion of the physical interpretation of the wave
functions will be found in Chapter XV.
13. THE FREE PARTICLE
A particle of mass m moving in a fieldfree space provides the
simplest application of the Schrodinger equation in three dimen
sions. Since V is constant (we choose the value zero for con
venience), the amplitude equation 128 assumes the following
form:
W + ^Wi = 0, (131)
or, in Cartesian coordinates,
a* + W i+ & + "»~ W * = ( }
This is a partial differential equation in three independent
variables x, y, and z. In order to solve such an equation it is
usually necessary to obtain three total differential equations,
one in each of the three variables, using the method of separation
of variables which we have already employed to solve the
Schrodinger time equation (Sec. 96). We first investigate the
possibility that a solution may be written in the form
Mx, y, z) = X(x) • Y(y) • Z(z), (133)
where X{x) is a function of x alone, Y(y) a function of y alone,
and Z(z) a function of z alone. If we substitute this expression
in Equation 132, we obtain, after dividing through by i/s the
equation
ld " X + iS + iS+^0. (.34)
X dx 2 ^ Y dy* ' Z dz* ' h 2
Since X is a function only of x, the first term does not change
its value when y and z change. Likewise the second term is
independent of x and z and the third term of x and y. Never
theless, the sum of these three terms must be equal to the con
stant jt"W f° r an y c h° ice of x, y, z. By holding y and z
IV13] THE FREE PARTICLE 91
fixed and varying x, only the first term can vary, since the others
do not depend upon x. However, since the sum of all the
terms is equal to a constant, we are led to the conclusion that
1 d 2 X
v ~J~2 * s independent of x as well as of y and z, and is therefore
itself equal to a constant. Applying an identical argument to
the other terms, we obtain the three ordinary differential
equations
I^fc 1*1 k and !*?* (135)
X dx* ~ Kx ' Y dy* Kv ' a Q Z dz 1 ~ *" K16 d)
with the condition
k x + h, + k, = ^jrW. (136)
It is convenient to put k x = r^J^x, which gives the equation
in z the form
d? + 1^ Z = ° (13_7)
This is now a total differential equation, which can be solved
by familiar methods. As may be verified by insertion in the
equation, a solution is
X(x) = N x sin <^V2mW x (x  xo)Y (138)
Since it contains two independent arbitrary constants N x and x 0j
it is the general solution. It is seen that the constant x defines
the location of the zeros of the sine function. The equations for
Y and Z are exactly analogous to Equation 137, and have the
solutions
Y(y) = N y sin ^V2mW w (y  y<>)}> J
Z(z) = N 8 sin <jV2mW z (z  z ) }•
(139)
The fact that we have been able to obtain the functions
X, Y, and Z justifies the assumption inherent in Equation 133.
It can also be proved 1 that no other solutions satisfying the
1 The necessary theorems are given in R. Courant and D. Hilbert,
"Methoden der mathematischen Physik," 2d ed., Julius Springer, Berlin,
1931.
92 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV13
boundary conditions can be found which are linearly independent
of these, i.e., which cannot be expressed as a linear combination
of these solutions.
The function \p must now be examined to see for what values of
W = W x + W y + W, it satisfies the conditions for an acceptable
wave function given in Section 9c. Since i^he sine function is
continuous, singlevalued, and finite for all real values of its
argument, the only restriction that is placed on W is that W x ,
W y , W» and therefore W be positive. We have thus reached the
conclusion that the free particle has a continuous spectrum of
allowed energy values, as might have been anticipated from the
argument of Section 9c.
The complete expression for the wave function corresponding
to the energy value
W = W x + W v + W, (1310)
is
Wz> V>z) = N sin \j\/ZrriW x {x — x )f
• sin ft V^mWviy  »o)} • sin <^V2mWz(z  zt>)\, (1311)
in which N is a normalization constant. The problem of the
normalization of wave functions of this type, the value of which
remains appreciable over an infinite volume of configuration
space (corresponding to a continuous spectrum of energy values),
is a complicated one. Inasmuch as we shall concentrate our
attention on problems of atomic and molecular structure, with
little mention of collision problems and other problems involving
free particles, we shall not discuss the question further, contenting
ourselves with reference to treatments in other books. 1
In discussing the physical interpretation of the wave functions
for this system, let us first consider that the physical situation is
represented by a wave function as given in Equation 1311
with W y and W z equal to zero and W x equal to W. The func
1 A. Sommerpeld, "Wave Mechanics," English translation by H. L.
Brose, pp. 293295, E. P. Dutton & Co., Inc., New York, 1929; Rxjark and
Urey, "Atoms, Molecules, and Quanta/' p. 541, McGrawHill Book Com
pany, Inc., New York, 1930.
IV13] THE FREE PARTICLE 93
tion 1 <&{x, y, z, t) = N sin \j\/2mW(x — Xo)/6 h is then a
set of standing waves with wave fronts normal to the x axis.
The wave length is seen to be given by the equation
x  v^f ' (13 " 12)
In classical mechanics the speed v of a free particle of mass m
moving with total energy W is given by the equation Y^mv 1 = W.
A further discussion of this system shows that a similar inter
pretation of W holds in the quantum mechanics. Introducing
v in place of W in Equation 1312, we obtain
X = — • (1313)
mv
This is the de Broglie expression 2 for the wave length associated
with a particle of mass m moving with speed v.
It is the sinusoidal nature of the wave functions for the free
particle and the similar nature of the wave functions for other
systems which has caused the name wave mechanics to be applied
to the theory of mechanics which forms the subject of this book.
This sinusoidal character of wave functions gives rise to experi
mental phenomena which are closely similar to those associated
in macroscopic fields with wave motions. Because of such
experiments, many writers have considered the wavelike char
acter of the electron to be more fundamental than its corpuscular
character, but we prefer to regard the electron as a particle and
to consider the wavelike properties as manifestations of the
sinusoidal nature of the associated wave functions. Neither
view is without logical difficulties, inasmuch as waves and
particles are macroscopic concepts which are difficult to apply to
microscopic phenomena. We shall, however, in discussing the
results of wavemechanical calculations, adhere to the particle
concept throughout, since we believe it is the simplest upon
which to base an intuitive feeling for the mathematical results
of wave mechanics.
1 It can be shown that the factors involving y and x in Equation 1311
approach a constant value in this limiting case.
1 L. db Broglib, Thesis, 1924; Ann. de phys. 3, 22 (1925).
94 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV13
The wave function which we have been discussing corresponds
to a particle moving along the x axis, inasmuch as a calculation
of the kinetic energy T x = k~pI associated with this motion
shows that the total energy of the system is kinetic energy of
motion in the x direction. This calculation is made by the
general method of Section 12d. The average value of T x is
 &f*'{M&* d '
= w x ,
or, since in this case we have assumed W x to equal W,
T x = W.
Similarly we find T r x = W r = (T x ) r , which shows, in accordance
with the discussion of Section 10c, that the kinetic energy of
motion along the x axis has the definite value W, its probability
distribution function vanishing except for this value.
On the other hand, the average value of p x itself is found on
calculation to be zero. The wave function
N sin \j\/2mW(x — x )fe ** h
hence cannot be interpreted as representing a particle in motion
in either the positive or negative direction along the x axis but
rather a particle in motion along the x axis in either direction,
the two directions of motion having equal probability.
The wave function N cos \j\/2mW(x — xo)fe r%h differs
from the sine function only in the phase, the energy being the
same. The sum and difference of this function and the sine func
tion with coefficient i are the complex functions
2Tt r= — si/ n 2riW. 2W /s — =.. . 2iriW \
^7^"^^ and N > e ~^ m ^*\— \
which are also solutions of the wave equation equivalent to the
sine and cosine functions. These complex wave functions
represent physical situations of the system in which the particle
is moving along the x axis in the positive direction with the
IV14] THE PARTICLE IN A BOX 95
definite momentum p x =» \/2mW or p x = — s/2mW, the
motion in the positive direction corresponding to the first of
the complex wave functions and in the negative direction to the
second. This is easily verified by calculation of pi and pj for
these wave functions.
The more general wave function of Equation 1311 also
represents a set of standing plane waves with wave length
X = h/\/2mW, the line normal to the wave fronts having the
direction cosines \/W x /W y \ZW y /W y and \ZW Z /W relative to the
x, y, and z axes.
Problem 131. Verify the statements of the next to the last paragraph
regarding the value of p x .
14. THE PARTICLE IN A BOX 1
Let us now consider a particle constrained to stay inside of a
rectangular box, with edges a, b> and c in length. We can repre
sent this system by saying that the potential function V(Xj y, z)
has the constant value zero within the region < x < a,
< y < by and < z < c y and that it increases suddenly in
value at the boundaries of this region, remaining infinitely
large everywhere outside of the boundaries. It will be found
that for this system the stationary states no longer correspond to
a continuous range of allowed energy values, but instead to a
discrete set, the values depending on the size and shape of the
box.
Let us represent a potential function of the type described as
V(x, y, z) = V x (x) + V v (y) + V,{z), (141)
the function V x (x) being equal to zero for < x < a and to
infinity for x < or x > a, and the functions V v (y) and V z {z)
showing a similar behavior. The wave equation
dV aV dV 8ir 2 m
dx 2 + dy 2 + dz 2 + h 2
{W  V x (x)  V y (y)  V&)}* = (142)
is separated by the same substitution
+(x, y } z) = X(x) • Y(y) • Z(z) (143)
1 Treated in Section Qd by the methods of the old quantum theory.
96 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV14
as for the free particle, giving three total differential equations,
that in x being
d*X , &r 2 m
dx*
+
{W x  V x (x)}X = 0.
(144)
In the region < x < a the general solution of the wave
equation is a sine function of arbitrary amplitude, frequency, and
phase, as for the free particle. Several such functions are repre
sented in Figure 141. All of these are not acceptable wave
w,
fr
o
w y
V x (x)
Fig. 141. — The potentialenergy function V x (x) and the behavior of X(x) near
the point x — a.
functions, however; instead only those sine functions whose
value falls to zero at the two points x = and x — a behave
properly at the boundaries. To show this, let us consider the
behavior of Curve A as x approaches and passes the value a,
using the type of argument of Section 9c. Curve A has a finite
positive value as x approaches a, and a finite slope. Its curvature
is given by the equation
d*X
dx 2
8* 2 m
A 2
\W X  V x {x))X.
(145)
At the point x = a the value of V(x) increases very rapidly and
without limit, so that, no matter how large a value the constant
W, has, Wx — V, becomes negative and of unbounded magni
IV141
THE PARTICLE IN A BOX
97
tude. The curvature or rate of change of the slope consequently
becomes extremely great, and the curve turns sharply upward
and experiences the infinity catastrophe. This can be avoided
in only one way; the function X(x) itself must have the value
zero at the point x = a, in order that it may then remain bounded
(and, in fact, have the value zero) for all larger values of x.
Similarly the sine function must fall to zero at x = 0, as shown
by Curve C. An acceptable wave function X(x) is hence a
sine function with a zero at x = and another zero at x = a,
Fig. 142. — The wave functions Xn x (x) and probability distribution functions
[Xn x (x)]* for the particle in a box.
thus having an integral number of loops in this region. The
phase and frequency (or wave length) are consequently fixed,
and the amplitude is determined by normalizing the wave func
tion to unity. Introducing the quantum number n x as the
number of loops in the region between and a, the wave length
becomes 2a/n x , and the normalized X(x) function is given by
the expression
X Vx (x) =
with
2 . n x TX
 sin ;
a a
n x = 1, 2, 3,
nnm 8ma 2
< x < a, (146)
(147)
98 WAVE EQUATION FOR A SYSTEM OF PARTICLES [VIH
The first four wave functions Xi(x), • • • , X 6 (x) are represented
in Figure 142, together with the corresponding probability
distribution functions { X n *(x) } 2 .
A similar treatment of the y and z equations leads to similar
expressions for Y nv (y) and Z n ,(z) and for W y and W t . The com
4
/
2H
x
116
2;
2
£
/
""
2q
2
IIS
^
2
2
^
£
/
^
2
/
114
^
z
^
^
2
2q
2:
/
113
^
z
2
z
£
/
^
^d
w?
2
2
^
121
l.M
n ~ M hi
/
/
n x
„ y ,, 2 * M,
^
^
s
s
^
s
* 211 ^jli_ —"411 511
%  l 4ll
r
«~ """"
0. . 'a
riojin
Fig. 143. — A geometrical representation of the energy levels for a particle in a
rectangular box.
plete wave function \p nx n v nXx, V, *) has the form, for values of x,
y, and z inside the box,
^nxn.n,^, 2/, 2) = A ^ sin sm —^ sin > (148)
\aoc .a b c
withn* = 1,2,3, • • • ;n y = 1,2,3, ■ • • ;n, = 1,2,3, • • • ;
and
WW ■= JF n , + TF„„ + W n , = ^ + Jf + £). (149)
The wave function ^ nxByn , can be described as consisting of
standing waves along the x, y, and z directions, with n x + 1
equally spaced nodal planes perpendicular to the x axis (begin
ning with x = and ending with x — a), n„ 4 1 nodal Dlanes
IV14]
THE PARTICLE IN A BOX
99
perpendicular to the y axis, and n t + 1 nodal planes perpendicular
to the z axis.
The various stationary states with their energy values may be
conveniently represented by means of a geometrical analogy.
Using a system of Cartesian coordinates, let us consider the
27
26
24
22
21
19
18
17
8ma
c 2 WJ
n x n y n z
•p=4 51le+c,333
•P =6 431 etc.
•p =3 422 e+c.
•P =3 332 e+c.
•P =6 421 e+c.
■P =3 33!e+c.
•P =3 4lle+c.
•P =3 322 e+c.
•P =6 321,132,213,312,231,123
• P = I 222
P =3 311,131,113
•P =3 221,122,212
• P =3 211.121,112
p = i Hi
Fia. 144.
Energy levels, degrees of degeneracy, and quantum numbers for a
particle in a cubic box.
lattice whose points have the coordinates n x /a y n v /b, and n t /c,
with n x = 1, 2, • • • ; n y = 1, 2, • • • ; and n ? = 1, 2 • • • .
This is the lattice defined in one octant about the origin by the
translations 1/a, 1/6, and 1/c, respectively; it divides the octant
into unit cells of volume 1/abc (Fig. 143). Each point of the
lattice represents a wave function. The corresponding energy
value is~
h 2
W = — 7 2
(1410)
100 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV15
in which l nx n>n. is the distance from the origin to the lattice point
n x nyn t , given by the equation
w. = vP+f+f ■ ( 14  n )
In case that no two of the edges of the box a, 6, and c are in
the ratio of integers, the energy levels corresponding to various
sets of values of the three quantum numbers are all different,
with one and only one wave function associated with each.
Energy levels of this type are said to be nondegenerate. If,
however, there exists an integral relation among a, 6, and c,
there will occur certain values of the energy corresponding to
two or more distinct sets of values of the three quantum numbers
and to two or more independent wave functions. Such an energy
level is said to be degenerate, and the corresponding state of the
system is called a degenerate state. For example, if the box is a
cube, with a = b = c, most of the energy levels will be degener
ate. The lowest level, with quantum numbers 111 (for n Xf
Uyj n z , respectively) is nondegenerate, with energy 3/i 2 /8ma 2 .
The next level, with quantum numbers 211, 121, and 112 and
energy 6/i 2 /8ma 2 , is triply degenerate. Successive levels, with
sets of quantum numbers and degrees of degeneracy (represented
by p), are shown in Figure 144. The degree of degeneracy
(the number of independent wave functions associated with a
given energy level) is often called the quantum weight of the
level.
15. THE THREEDIMENSIONAL HARMONIC OSCILLATOR IN
CARTESIAN COORDINATES
Another threedimensional problem which is soluble in Car
tesian coordinates is the threedipiensional harmonic oscillator,
a special case of which, the isotropic oscillator, we have treated
in Section la by the use of classical mechanics. The more general
system consists of a particle bound to the origin by a force whose
components along the x, y, and z axes are equal to — k x x, —Jc y y,
and — k t z, respectively, where k x> k y , k z are the force constants
in the three directions and x, y, z are the components of the
displacement along the three axes. The potential energy
is thus
V = HktZ* + y 2 k y y* + %k z z\ (15^1)
IV15] THREEDIMENSIONAL HARMONIC OSCILLATOR 101
which, on introducing instead of the constants k x> k v , k t their
expressions in terms of the classical frequencies v x , v v , v» } becomes
smce
V = 27r 2 m(v£c 2 + v\y 2 + v 2 z 2 ),
k x = 4ir 2 mvly
k y = 47r 2 mv 2
(162)
(153)
k, = 4ir 2 m*/ 2 .
The general wave equation 128 thus assumes for this problem
the form
dx 2
by 2
av
dz 2
^ + ^ + ^ + S ^W 2* 2 m(v 2 x> + v\y 2 + &*)) +
h 2
which, on introducing the abbreviations
&w 2 m rTr
W,
and
simplifies to the equation
h 2
a x
=
4w 2 m
h '
Oy
=
47r 2 m
h '
rt.
4r 2 m
1
o,
(164)
(155a)
(1556)
(165c)
(155d)
To solve this equation we proceed in exactly the same manner
as in the case of the free particle (Sec. 13) ; namely, we attempt to
separate variables by making the substitution
*(*, y, z) = X(x) • 7(2/) • Z(z). (157)
This gives, on substitution in Equation 156 and division of the
result by ^, the equation
(158)
102 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV15
It is evident that this equation has been separated into terms
each of which depends upon one variable only; each term is
therefore equal to a constant, by the argument used in Section 13.
We obtain in this way three total differential equations similar
to the following one :
^) + (x, _ a l x *)X(x) = 0, (159)
in which X* is a separation constant, such that
X* + X* + Xz = X. (1510)
Equation 159 is the same as the wave equation 112 for the
onedimensional harmonic oscillator which was solved in Section
11. Referring to that section, we find that X(x) is given by the
expression
X(x) = N n ,e 2 Hn (V&c) (1511)
and that X x is restricted by the relation
X, = (2n x + 1)«„ (1512)
in which the quantum number n x can assume the values 0, 1,
2, • • • . Exactly similar expressions hold for Y(y) and Z[z)
and for X^ and X*. The total energy is thus given by the equation
W nx n v n, = A { (n x + Y 2 ) V x + (lly + %) V y + (n z + %)».}, (1513)
and the complete wave function by the expression
tn z n v nXx> V, «) =
^.n.*/^^"^ (1514)
The normalizing factor has the value
_ J (ojw)" V* n*i»«
For the special case of the isotropic oscillator, in which
v x = v y = v, = vo and a x = oty = a e , Equation 1513 for the energy
reduces to the form
W = (n x + ny + n z + %)hv  (n + %)hv . (1516)
n = n x + riy + n z may be called the total quantum number.
Since the energy for this system depends only on the sum of the
IV16]
CURVILINEAR COORDINATES
103
quantum numbers, all the energy levels for the isotropic oscilla
tor, except the lowest one, are degenerate, with the quantum
weight ~ lyLJL—L. Figure 151 shows the first few energy
II
n»4
— Oslt
1
9
n3
 p10
300,030,003,
210,011 JOZ,
2
201,120,012,
1/1. '
W 7
n = 2
 d«6 $200,020,002,
bv >
Z
p *\iio',ioi',on.
5
n»l
wmm n»^ //9/7 /)//) /)/0/
2
3
n«0
2
Fig. 151.
Energy levels, degrees of degeneracy, and quantum numbers for the
threedimensional isotropic harmonic oscillator.
levels, together with their quantum weights and quantum
numbers.
16. CURVILINEAR COORDINATES
In Chapter I we found that curvilinear coordinates, such as
spherical polar coordinates, are more suitable than Cartesian
coordinates for the solution of many problems of classical
mechanics. In the applications of wave mechanics, also, it is
very frequently necessary to use different kinds of coordinates.
In Sections 13 and 15 we have discussed two different systems,
the free particle and the threedimensional harmonic oscillator,
whose wave equations are separable in Cartesian coordinates.
Most problems cannot be treated in this manner, however, since
it is usually found to be impossible to separate the equation into
three parts, each of which is a function of one Cartesian coordi
nate only. In such cases there may exist other coordinate
systems in terms of which the wave equation is separable, so
that by first transforming the differential equation into the proper
104 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV16
coordinates the same technique of solution may often be
applied.
In order to make such a transformation, which may be repre
sented by the transformation equations
x = f(u, v, w), (16la)
V = g(u, v, w), (1616)
z = h(u, v f w), (16lc)
it is necessary to know what form the Laplace operator V 2 assumes
in the new system, since this operator has been defined only in
Cartesian coordinates by the expression
d 2 d 2 d 2
*'  1? + £■ + y (i6  2)
The process of transforming these second partial derivatives is a
straightforward application of the principles of the theory of
partial derivatives and leads to the result that the operator V 2
in the orthogonal coordinate system uvw has the form
V2 = 1 j d (q v q w d\ 6 /q u q w d\ ±(qyqv _d\)
quq v qw\du\ q u bu) dv\ q v dvj dw\ q w dw/j
in which
(163)
Equation 163 is restricted to coordinates u, v, w which are
orthogonal, that is, for which the coordinate surfaces represented
by the equations u = constant, v = constant, and w = constant
intersect at right angles. All the common systems are of this
type.
The volume element dr for a coordinate system of this type
is also determined when q u , q v , and q w are known. It is given by
the expression
dr = dxdydz = q u q v q w dudvdw. (165)
In Appendix IV, q u , q v , qw, and v 2 itself are given for a number of
important coordinate systems.
IV17] THREEDIMENSIONAL HARMONIC OSCILLATOR 105
Mathematicians 1 have studied the conditions under which the
wave equation is separable, obtaining the result that the three
dimensional wave equation can be separated only in a limited
number of coordinate systems (listed in Appendix IV) and then
only if the potential energy is of the form
V = qu$u(u) + q v $v(v) + q w <$>w(w),
in which 4> w (w) is a function of u alone, $ v (v) of v alone, and
$ w (w) of w alone.
17. THE THREEDIMENSIONAL HARMONIC OSCILLATOR IN
CYLINDRICAL COORDINATES
The isotropic harmonic oscillator in space is soluble by separa
tion of variables in several coordinate systems, including Car
..**
((>*%&
*Y
Fia. 171. — Diagram showing cylindrical coordinates.
tesian, cylindrical polar, and spherical polar coordinates. We
shall use the cylindrical system in this section, comparing the
results with those obtained in Section 15 with Cartesian
coordinates.
Cylindrical polar coordinates p, <p, z, which are shown in'
Figure 171, are related to Cartesian coordinates by the equations
of transformation
x = p cos <p,\
y = psin ip}
z = z. \
(171)
1 H. P. Robertson, Mathematische Annalen 98, 749 (1928); L. P. Eisbn
hart, Ann. Mathematics 35, 284 (1034).
106 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV17
Reference to Appendix IV shows that V 2 in terms of p, <p, z has
the form
Vi.i i/p^ + I^+*!. (172)
p dp\ p d P J ^ p 2 d<p 2 ^ dz* u ;
Consequently, the wave equation 154 for the threedimensional
harmonic oscillator becomes
p dp\dpj p
dhp 6V 87r 2 m
1 ^o I
p 2 dv> 2 az 2 T h 2
{W  Wm{y\p 2 + viz 2 )}* = 0, (173)
when we make v x = v y = v (only in this case is the wave equa
tion separable in these coordinates). Making the substitutions
x 87r 2 m T1T /ihT , ,
X = rrW, (174a)
and
we obtain the equation
a = —rPo, (1746)
a* = — ry., (174c)
p ^VW + ? a7 2 + a* 2 + (x "" ap  °*^ = a (17 " 5)
Pursuing the method used in Section 15, we try the substitution
* = P( P ) • *(*) • Z(*), (176)
in which P(p) depends only on p, <!>(<?) only on <p> and Z(z) only
on z. Introduction of this into Equation 175 and division by \p
leads to the expression
1 d/dP\ 1 d 2 <i> ld 2 Z 22
Fp^VW + ^d? + z^ + x ^ ap a ** =a (17 " 7)
The terms of this equation may be divided into two classes:
those which depend only on z and those which depend only on
p and <p. As before, since the two parts of the equation are func
tions of different sets of variables and since their sum is constant,
IV17] THREEDIMENSIONAL HARMONIC OSCILLATOR 107
each of the two parts must be constant. Therefore, we obtain
two equations
Jf + (A,  a*,z*)Z = (178)
dz
and
with
rpdh%) + 7*d? af > t + x = ' (17 ~ 9)
X' + X, = X
The first of these is the familiar onedimensional harmonic
oscillator equation whose solutions
Z n ,(z) = N nz e~^H ni (V^zz) (1710)
are the Hermite orthogonal functions discussed in Section lie.
As in the onedimensional problem, the requirement that the
wave function satisfy the conditions of Section 9c restricts the
parameter X* to the values
X* = (2n z + l)o., n z = 0, 1, 2, • • • . (1711)
Equation 179, the second part of the wave equation, is a
function of p and v and so must be further separated. This
may be accomplished by multiplying through by p 2 . The
second term of the resulting equation is independent of p; it is
therefore equal to a constant, which we shall call —m 2 . The
two equations we obtain are the following:
d 2 *
and
The first of these is a familiar equation whose normalized solution
is 1
<!>(<?) = — 7= e imip . (1714)
Inasmuch as e im * is equal to cos trap + i sin m<p, we see that for
arbitrary values of the separation constant m this function is
1 Instead of the exponential, the forms &(<p) = N cos m<p and N sin nap
may be used. See Section 186, Chapter V.
108 WAVE EQUATION FOR A SYSTEM OF PARTICLES (IV17
not single valued; that is, <£ does not have the same value for
if> = and for <p = 2r, which correspond to the same point in
space. Only when m is a positive or negative integer or zero is
<£ singlevalued, as is required in order that it be an acceptable
wave function (Sec. 9c) ; m must therefore be restricted to such
values. <p is called a cyclic coordinate (or ignorable coordinate),
these names being applied to a variable which does not occur
anywhere in the wave equation (although derivatives with respect
to it do appear). Such a coordinate always enters the wave
function as an exponential factor of the type given in Equation
1714. *
The equation for P(p) may be treated by the same general
method as was employed for the equation of the linear harmonic
oscillator in Section 11a. The first step is to obtain an asymp
totic solution for large values of p, in which region Equation
1713 becomes approximately
~  <* 2 p 2 P = 0. (1715)
The asymptotic solution of this is e 2 , since this function
satisfies the equation
which reduces to 1715 for large values of p. Following the
reasoning of Section 11a, we make the substitution
P(p) = e~^/(p) (1716)
in Equation 1713. From this we find that / must satisfy the
equation
1 m 2
f"  2a P f + if + (X'  2a)/  ^f = 0. (1717)
As before, it is convenient to replace p by the variable
*  V^P (1718)
and/(p) by F(£), a process which gives the equation
% *5HS +£ 2 f>= ' <»*>
1 Condon and Morse, "Quantum Mechanics," p. 72.
IV17] THREEDIMENSIONAL HARMONIC OSCILLATOR 109
We could expand F directly as a power series in £, as in Section
11a. This is not very convenient, however, because the first
few coefficients would turn out to be zero. Instead, we make
the substitution
00
F(S) = ?X a '? = a °*' + ai *' +1 + ' ' ' ' (17 " 20)
in which s is an undetermined parameter and a is not equal
to zero.
This substitution is, indeed, called for by the character of the differential
equation. 1 Equation 1719 is written in the standard form
d*F dF
^p + Pte^ + tfero.
d*F
the coefficient of — — being unity. The coefficients p and q in Equation
dp
1719 possess singularities 2 at £ = 0. The singular point £ = is a regular
point, however, inasmuch as p(£) is of order l/£ and q(£) of order l/£ 2 . To
solve a differential equation possessing a regular point at the origin, the
substitution 1720 is made in general. It is found that it leads to an indidal
equation from which the index s can be determined.
Since we are interested only in acceptable wave functions, we shall ignore
negative values of s. For this reason we could assume F(£) to contain only
positive powers of £. Occasionally, however, the indicial equation leads to
nonintegral values of s, in which case the treatment is greatly simplified by
the substitution 1720.
If we introduce the series 1720 into Equation 1719 and group
together coefficients of equal powers of £ , we obtain the equation
(s 2  m 2 )a e~ 2 + {(* + l) 2  w^oif 1
+
+
{(« + 2) 2  m 2 \a 2 + \~~ 2( 5 + l)a ]f + • • •
{(« + v) 2  m 2 )a v + <X  2(« + v  1)L_ 2 1^"
+ . . . =o. (1721)
Since this is an identity in £ , that is, an equation which is true
for all values of £ , we can show that the coefficient of each power
1 See the standard treatments of the theory of linear differential equations;
for example, Whittaker and Watson, "Modern Analysis," Chap. X.
* A singularity for a function p(£) is a point at which p(£) becomes infinite.
110 WAVE EQUATION FOR A SYSTEM OF PARTICLES [IV17
of £ must be itself equal to zero. This argument gives the set of
equations
(s 2  m 2 )a = 0, (1722a)
{(« + l) 2  m 2 }^ = 0, (17226)
(1722c)
{(s + v) 2  m*}a v + $L  2(* + v  1) ja,_ 2 = 0,
etc.
The first of these, 1722a, is the indicial equation. From it
we see that s is equal to +m or — ra, inasmuch as a is not equal
to zero. In order to obtain a solution of the form of Equation
1720 which is finite at the origin, we must have s positive, so
that we choose s = +\m\. This value of s inserted in Equation
17226 leads to the conclusion that a,i must be zero. Since the
general recursion relation 1722c connects coefficients whose
subscripts differ by two, and since a\ is zero, all odd coefficients
are zero. The even coefficients may be obtained in terms of
a by the use of 1722c.
However, just as in the case of the linear harmonic oscillator,
the infinite series so obtained is not a satisfactory wave function
for general values of X', because its value increases so rapidly with
increasing £ as to cause the total wave function to become
infinite as £ increases without limit. In order to secure an
acceptable wave function it is necessary to cause the series to
break off after a finite number of terms. The condition that the
series break off at the term a n 'Z n ' Hm] , where n f is an even integer,
is obtained from 1722c by putting n' + 2 in place of v and equat
ing the coefficient of a n , to zero. This yields the result
V = 2(m h n! + 1)«. (1723)
Combining the expressions for \ z and X' given by Equations
1711 and 1723, we obtain the result
X = V + X, = 2(m + ri + l)a + 2(n, + H)a z , (1724)
or, on insertion of the expressions for X, a, and a z ,
W mn >n t = (H + n' + l)hv + (n. + \Qhv t . (1725)
In the case of the isotropic harmonic oscillator, with v z = vo»
this becomes
IV17] THREEDIMENSIONAL HARMONIC OSCILLATOR 111
W n = (n + y 2 )hv Qi n = \m\+n' + n.. (1726)
The quantum numbers are restricted as follows:
m = 0, ±1, ±2, • • • ,
n' = 0, 2, 4, 6, • • ,
n* = 0, 1, 2, • • • .
These lead to the same quantum weights for the energy levels
as found in Section 15.
The wave functions have the form
$n>mn(p, <P, z) = Ne^e r F m , ,»' ( V«p)e 2 * H n ,(V<**),
(1727)
in which JV is the normalization constant and F, m ,, n >(\/ap) is a
polynomial in p obtained from Equation 1720 by the use of the
recursion relations 1722 for the coefficients a v . "It contains only
odd powers of p if \m\ is odd, and only even powers if \m\ is
even.
Problem 171. The equation for the free particle is separable in many
coordinate systems. Using cylindrical polar coordinates, set up and
separate the wave equation, obtain the solutions in <p and z, and obtain the
recursion formula for the coefficients in the series solution of the p equation.
Hint: In applying the polynomial method, omit the step of finding the
asymptotic solution.
Problem 172. Calculate pi for a harmonic oscillator in a state repre
sented by ypn'mn, of Equation 1727. Shew that p x is zero in the same state.
Hint: Transform — into cylindrical polar coordinates.
Problem 173. The equation for the isotropic harmonic oscillator is
separable also in spherical polar coordinates. Set up the equation in these
coordinates and carry out the separation of variables, obtaining the three
total differential equations.
CHAPTER V
THE HYDROGEN ATOM
The problem of the structure of the hydrogen atom is the most
important problem in the field of atomic and molecular structure,
not only because the theoretical treatment of this atom is simpler
than that of other atoms and of molecules, but also because it
forms the basis for the discussion of more complex atomic sys
tems. The wavemechanical treatment of polyelectronic atoms
and of molecules is usually closely related in procedure to that
of the hydrogen atom, often being based on the use of hydrogen
like or closely related wave functions. Moreover, almost without
exception the applications of qualitative and semiquantitative
wavemechanical arguments to chemistry involve the functions
which occur in the treatment of the hydrogen atom.
The hydrogen atom has held a prominent place in the develop
ment of physical theory. The first spectral series expressed by a
simple formula was the Balmer series of hydrogen. Bohr's
treatment of the hydrogen atom marked the beginning of the old
quantum theory of atomic structure, and wave mechanics had
its inception in Schrodinger's first paper, in which he gave the
solution of the wave equation for the hydrogen atom. Only
in Heisenberg's quantum mechanics was there extensive develop
ment of the theory (by Heisenberg, Born, and Jordan) before
the treatment of the hydrogen atom, characterized by its diffi
culty, was finally given by Pauli. In later developments, beyond
the scope of this book, the hydrogen atom retains its important
position; Dirac's relativistic quantum theory of the electron
is applicable only to oneelectron systems, its extension to
more complicated systems not yet having been made.
The discussion of the hydrogen atom given in this chapter is
due to Sommerfeld, differing in certain minor details from that
of Schrddinger. It is divided into four sections. In the first,
Section 18, the wave equation is separated and solved by the
polynomial method, and the energy levels are discussed. Sec
112
V18a] THE SOLUTION OF THE WAVE EQUATION 113
tions 19 and 20 include the definition of certain functions, the
Legendre and Laguerre functions, which occur in the hydrogen
atom wave functions, and the discussion of their properties. A
detailed description of the wave functions themselves is given
in Section 21.
18. THE SOLUTION OF THE WAVE EQUATION BY THE POLY
NOMIAL METHOD AND THE DETERMINATION OF
THE ENERGY LEVELS
18a. The Separation of the Wave Equation. The Transla
tional Motion. — We consider the hydrogen atom as a system of
two interacting point particles, the interaction being that due
to the Coulomb attraction of their electrical charges. Let us
for generality ascribe to the nucleus the charge +Ze, the charge
of the electron being —e. The potential energy of the system,
Ze 2
in the absence of external fields, is > in which r is the distance
r
between the electron and the nucleus.
If we write for the Cartesian coordinates of the nucleus and
the electron x\ } yi, Z\ and x^ y^ £2, and for their masses mi and
ra 2 , respectively, the wave equation has the form
1 (d 2 ^ <fth ayA , ±_(<*ti . *Vj . a yA
m\ dx\ + dy\ + dz\ ) + m\ dxl + dy\ + dz\ )
fcn2
+ jfiWr ~ V)+t = 0, (181)
in which
V = 
Ze 2
V(X2  *i) 2 + (2/2  Vi) 2 + (Z2  zi) 2
Here the subscript T (signifying total) is written for W and ^ to
indicate that these quantities refer to the complete system, with
six coordinates.
This equation can be immediately separated into two, one of
which represents the translational motion of the molecule as a
whole and the other the relative motion of the two particles.
In fact, this separation can be accomplished in a somewhat more
general case, namely, when the potential energy V is a general
function of the relative positions of the two particles, that is,
V = V(x*i — x\ 9 2/2 — yi, z*k — 21). This includes, for example,
the hydrogen atom in a constant electric field, the potential
114 THE HYDROGEN ATOM [V18a
energy due to the field then being eEz* — eEzi = eE(z* — Z\) f
in which E is the strength of the field, considered as being in the
direction of the z axis.
To effect the separation, we introduce the new variables
x, t/, and z, which are the Cartesian coordinates of the center of
mass of the system, and r, #, and <p, the polar coordinates of the
second particle relative to the first. These coordinates are
related to the Cartesian coordinates of the two particles by the
equations
X  . >
rrii + m 2
(182a)
rriiyi + m 2 y 2
y — ,
mi + m 2
(1826)
miZi + ra 2 2 2
z = . y
mi + m 2
(182c)
r sin # cos <p = x 2 — xi,
(182d)
r sin & sin ^> = y 2 — yi,
(1826)
r cos t? = 2 2 — 2i.
(182/)
The introduction of these new independent variables in
Equation 181 is easily made in the usual way. The resultant
wave equation is
l / ay r ay r ay r \ in_ ^ / ,d*A
mi + m 2 \^ ax 2 "*" dy 2 + a* 2 j "*" M r 2 dry ar /
l aVr
j : — r_m +
• ~2 «i^2 JH 3.2 »
(**£)}
r 2 sin 2 # d<p 2 ' r 2 sin # d&
&7T 2
+ ^ { W T  V(r, », <p)\+t= 0. (183)
In this equation the symbol /x has been introduced to represent
the quantity
raira 2
mi + m 2
fori = — +— V (184)
y ju mi m 2 /
M is the reduced mass of the system, already discussed in Section
2d in the classical treatment of this problem.
It will be noticed that the quantity in the first set of parentheses
is the Laplacian of \f/ T in the Cartesian coordinates x, y } and 2,
and the quantity in the first set of braces is the Laplacian in the
polar coordinates r, #, and <p (Appendix IV).
V18a] THE SOLUTION OF THE WAVE EQUATION 115
We now attempt to separate this equation by expressing xpr
as the product of a function of x> y, z and a function of
r, #, <p, writing
iMx, y, z, r, tf, <p) = F(z, y, z)+(r, *, *). (185)
On introducing this in Equation 183 and dividing through by
\p T = Fyf/j it is found that the equation is the sum of two parts,
one of which is dependent only on x f y, and z and the other
only on r, #, and (p. Each part must hence be equal to a con
stant. The resulting equations are
d 2 F d 2 F d 2 F 8 T 2( mi +m 2 )
W + W + W + h 2 WtrF = °> (18_6)
and
r 2 dry dr)"^ r 2 sin 2 t> <V "*" r 2 sin d0^ Sin d#/
+ ^{Tf  7(r, #,*)}*=(>, (187)
with
TT , r + w = Wt. (188)
Equation 186 is identical with Equation 132 of Section 13,
representing the motion of a free particle; hence the translational
motion of the system is the same as that of a particle with mass
mi + m 2 equal to the sum of the masses of the two particles.
In most problems the state of translational motion is not impor
tant, and a knowledge of the translational energy W tr is not
required. In our further discussion we shall refer to W, the
energy of the system aside from the translational energy, simply
as the energy ctf the system.
Equation 187 is identical with the wave equation of a single
particle of mass m under the influence of a potential function.
V(r f #, if) . This identity corresponds to the classical identity of
Section 2d (Eqs. 225).
If we now restrict ourselves to the case in which the potential
function V is a function of r alone,
V = V(r),
Equation 187 can be further separated. We write
f (r, *, <p) = R(r) ■ 9(#) • #(*) ; (189)
116 THE HYDROGEN ATOM [V18a
on introducing this in Equation 187 and dividing by RQ$, it
becomes
JL il 2^\ . 1 d 2 * 1 dj . d§\
r 2 R dr\ dr) + > 2 sin 2 #<*> d<p 2 "*" r 2 sin #0 d#\ Sm dt>/
+ %r\w V(r)} o. (1810)
On multiplying through by r 2 sin 2 #, the remaining part of the
1 d 2 $
second term, — ^—z, which could only be a function of the inde
<f> d<p 2
pendent variable <p, is seen to be equal to terms independent of <p.
Hence this term must be equal to a constant, which we call — m 2 :
d 2 $
The equation in & and r then can be written as
1 l( r ^\ _ J** _L_ A( 8in <M + §^
R dr\ dr / sin 2 ^ sin dQ d&\ d#J ^ h
{W  V(r)\ = 0.
The part of this equation containing the second and third terms
is independent of r and the remaining part is independent of #,
so that we can equate each to a constant. If we set the # terms
equal to the constant — 0, and the r terms equal to +fi f we
obtain the following equations, after multiplication by 9 and
by R/r 2 , respectively:
8irV 2
2
and
Equations 1811, 1812, and 1813 are now to be solved
in order to determine the allowed values of the energy.
The sequence of solution is the following: We first find that
Equation 1811 possesses acceptable solutions only for certain
values of the parameter m. Introducing these in Equation 1812,
we find that it then possesses acceptable solutions only for
certain values of 0. Finally, we introduce these values of
V18bl THE SOLUTION OF THE WAVE EQUATION 117
in Equation 1813 and find that this equation then possesses
acceptable solutions only for certain values of W. These are the
values of the energy for the stationary states of the system.
It may be mentioned that the wave equation for the hydrogen
atom can also be separated in coordinate systems other than the
polar coordinates r, #, and <p which we have chosen, and for some
purposes another coordinate system may be especially appro
priate, as, for example, in the treatment of the Stark effect,
for which (as shown by Schrodinger in his third paper) it is
convenient to use parabolic coordinates.
18b. The Solution of the <p Equation. — As was discussed in
Section 17, the solutions of Equation 1811, involving the cyclic
coordinate <p, are
<M*>) = *=^*. (1814)
In order for the function to be singlevalued at the point <p —
(which is identical with <p = 2tt)> the parameter m must be equal
to an integer. The independent acceptable solutions of the (p
equation are hence given by Equation 1814, with m = 0,
+ 1, +2, • • • , — 1, ^2, • • k ,* these values are usually
written as 0, ±1, ±2, • • • , it being understood that positive
and negative values correspond to distinct solutions.
The constant m is called the magnetic quantum number. It is
the analogue of the same quantum number in the oldquantum
theory treatment (Sec. 76).
The factor 1/\/2t is introduced in order to normalize the
functions $m(<p), which then satisfy the equation
J%*M<M*>)^ = 1. (1815)
It may be pointed out that for a given value of \m\ (the
absolute value of ra), the two functions $\ m \(<p) and 3>_ w (¥?)
satisfy the same differential equation, with the same value of the
parameter, and that any linear combination of them also satisfies
the equation. The sum and the difference of these two functions
are the cosine and sine functions. It is sometimes convenient
to use these in place of the complex exponential functions as the
independent solutions of the wave equation, the normalized
solutions then being
118 THE HYDROGEN ATOM [V18c
1
$o(v>) =
V3ftr
/ 1
cos \m\ <p, \ (1816)
$\m\(<p) =
V2t
\\Zw
— /sin \m\<p, \m\ = 1,2,3, • •
There is only one solution for \m\ = 0. These functions are
normalized and are mutually orthogonal.
It is sometimes convenient to use the symbol m to represent
the absolute value of the magnetic quantum number as well
as the quantum number itself. To avoid confusion, however,
we shall not adopt this practice but shall write ra for the
absolute value of m.
18c. The Solution of the # Equation. — In order to solve the
& equation 1812, it is convenient for us to introduce the new
independent variable
z = cos tf, (1817)
which varies between the limits — 1 and +1, and at the same time
to replace 9(#) by the function P(z) to which it is equal:
P(z) = 6(#). (1818)
Noting that sin 2 # = 1 — z 2 and that
d9 dPdz^ _dP .
d* dz dd dz Sm '
we see that our equation becomes
On attempting to solve this equation by the polynomial method,
it is found that the recursion formula involves more than two
terms. If, however, a suitable substitution is made, the equa
tion can be reduced to one to which the polynomial method can
be applied.
The equation has singular points at z = ±1, both of which are regular
points (see Sec. 17), so that it is necessary to discuss the indicial equation
at each of these points. In order to study the behavior near z — f1, it L?
V18c] THE SOLUTION OF THE WAVE EQUATION 119
convenient to make the substitution x  1 — z, R(x) = P(z), bringing this
point to the origin of x. The resulting equation is
00
If we substitute R — x* ^X a^x" in this equation, we find that the indicial
equation (see Sec. 17) leads to the value \m\/2 for s. Likewise, if we investi
gate the point z = —1 by making the substitution y = 1 } z and similarly
study the indicial equation at the origin of y, we find the same value for the
index there.
The result of these considerations is that the substitution
I ml 1ml I ml
P(z) = x2y2G(z) = (1  z 2 ) YG(z) (1820)
is required. On introducing this into Equation 1819, the differ
ential equation satisfied by G(z) — which should now be directly
soluble by a power series — is found to be
(1  z 2 )G"  2(m + l)zG' +
[p  \m\(\m\ + 1))0 = 0, (1821)
in which G' represents r and G" represents rj
This equation we now treat by the polynomial method, the
successive steps being similar to those taken in Section 11 in the
discussion of the harmonic oscillator. Let
G = a + aiz + a 2 z 2 + a z z 3 + • • • , (1822)
with G' and G" similar series obtained from this by differentiation.
On the introduction of these in Equation 1821, it becomes
1 • 2a 2 + 2 • 3a 3 z + 3 • 4a 4 z 2 + 4 • 5a b z* + • • •
 1 • 2a 2 z 2  2 • 3a 3 z 3  • • •
2(m + l)aiz 2 • 2(m + l)a 2 z 2 2 • 3(m + l)a 3 z 3  • • •
+ {P  m(m + l)}a + {}a l2 + {}a 2 z 2 + {}a 3 z 3 + • • • = 0,
in which the braces {} represent {/3 — m(m + 1)}. This
equation is an identity in z, and hence the coefficients of indi
vidual powers of z must vanish; that is,
1 • 2a 2 + {}a = 0,
23a 3 + ({)  2(m + l)) fll =0,
3 • 4a 4 + ({ }  2 • 2(m + 1)  1 . 2)a 2 = 0,
4 • 5a 6 + ({ }  2 • 3(m + 1)  2 • 3)a 8 = 0,
120 THE HYDROGEN ATOM [V18c
or, in general, for the coefficient of z",
(v + 1)(» + 2K+ 2 + UP  \m\(\m\ + 1)}
 2v(\m\ + 1)  v(v  1)K = 0.
This leads to the twoterm recursion formula
_ JL± H)(" + H + 1 ) 0 „
(„ + !)(„ + 2)
(1823)
between the coefficients a„ +2 and a, in the series for (?.
It is found on discussion by the usual methods 1 that an infinite
series with this relation between alternate coefficients converges
(for any values of \m\ and p) for — 1 < z < 1, but diverges for
z = +1 or — 1, and in consequence does not correspond to an
acceptable wave function. In order to be satisfactory, then,
our series for G must contain only a finite number of terms.
Either the even or the odd series can be broken off at the term
in z v ' by placing
P = (/ + \m\)(v' + \m\ + 1), / = 0, 1, 2, • • • ,
and the other series can be made to vanish by equating a\ or a
to zero. The characteristic values of the parameter p are thus
found to be given by the above expression, the corresponding
functions G(z) containing only even or odd powers of z as /
is even or odd.
It is convenient to introduce the new quantum number
I = v' + \m\ (1824)
in place of /, the allowed values for I being (from its definition)
m, \m\ + 1, m + 2, • • • . The characteristic values of p
are then
P = 1(1 + 1), I = (m, \m\ + 1, ' * ' . (1825)
I is called the azimuthal quantum number; it is analogous to the
quantum number k of the old quantum theory. Spectral states
which are now represented by a given value of I were formerly
represented by a value of k one unit greater, k = 1 corresponding
to I = 0, and so on.
1 R. Courant and D. Hilbert, " Methoden der mathematischen Physik,"
2d ed.,Vol. I, p. 281, Julius Springer, Berlin, 1931.
V18d]
THE SOLUTION OF THE WAVE EQUATION
121
We have now shown that the allowed solutions of the # equa*
tion are 9(#) = (1  2 2 ) lm/2 G(z), in which G(z) is denned by
the recursion formula 1823, with p = 1(1 + 1). It will be shown
in Section 19 that the functions G(#) are the associated Legendre
functions. A description of the functions will be given in
Section 21.
18d. The Solution of the r Equation. — Having evaluated /? as
1(1 + 1), the equation in r becomes
i d( dR\ r Ki + D
r 2 dr\ dr J ^ _ r 2 ^
S ^ {W _ v(r)\
R = 0, (1826)
in which V(r) = — Ze 2 /r, Z being the atomic number of the atom.
It is only now, by the introduction of this expression for the
potential energy, that we specialize the problem to that of the
oneelectron or hydrogenlike atom. The discussion up to this
point is applicable to any system of two particles which interact
with one another in a way expressible by a potential function
V(r)j as, for example, the two nuclei in a diatomic molecule after
the electronic interactions have been considered by the Born
Oppenheimer method (Sec. 35a).
Let us first consider the case of W negative, corresponding to
a total energy insufficient to ionize the atom. Introducing the
symbols
and
8t 2 ixW \
h 2
±T 2 >iZe 2
h 2 a '
(1827)
and the new independent variable
p = 2ar,
the wave equation becomes
1 d( dS\ ( 1 1(1 + i) , x\
(1828)
^ p ^ » , (1829)
122 THE HYDROGEN ATOM [VlSd
in which S(p) = R (r). As in the treatment of the harmonic
oscillator, we first discuss the asymptotic equation. For p
large, the equation approaches the form
dp
the solutions of which are
d*S l e
S = e^ and 8 = e \
Only the second of these is satisfactory as a wave function.
We now assume that the solution of the complete equation
1829 has the form
p
S(p) = e 2 F(p). (1830)
The equation satisfied by F(p) is found to be
^ p ^ oo. (1831)
The coefficients of F' and F possess singularities at the origin,
which is a regular point (cf. Sec. 17), so that we again make the
substitution
F(p) = P 'L(p), (1832)
in which L(p) is a power series in p beginning with a non vanishing
constant term:
L(p) = %a»P v , a 5* 0. (1833)
V
Since
F'(p) = sp''L + p'U
and
F"( P ) = s(s  1) P  2 L + 2s P  1 L' + p'L",
Equation 1831 becomes
p+*L" + 2sp'+ 1 L' + s(s  l)p'L
+ 2p'+ l L' + 2sp'L
 p«+ 2 L'  sp'+ l L
+ (X  1) P '+>L  1(1 + l)p'L = 0. (1834)
V18d] THE SOLUTION OF THE WAVE EQUATION 123
Since L begins with the term a , the coefficient of p* is seen to be
{s(s — 1) + 2s — 1(1 + l)}a , and, since a does not vanish,
the expression in braces must vanish in order for Equation 1834
to be satisfied as an identity in p. This gives as the indicial
equation for s:
s(s + 1)  1(1 + 1) = 0, or 5 = +1 or (1 + 1). (1835)
Of the two solutions of the indicial equation, the solution
s = — (J + 1) does not lead to an acceptable wave function.
We accordingly write
F(p) = p l L(p), (1836)
and obtain from 1834 the equation
P L" + {2(1 + 1)  p}Z/ + (X  I  1)L = 0, (1837)
after substituting I for s and dividing by p l+l . We now introduce
the series 1833 for L in this equation and obtain an equation
involving powers of p, the coefficients of which must vanish
individually. These conditions are successively
(X  I  l)a + 2(1 + l) ai = 0,
(X — I — 1 — IK + {2 . 2(i + 1) + 1 • 2}a 2 = 0,
(X  I  1  2)a 2 + {32(Z + 1) +23}a 3 =
or, for the coefficient of p",
(X — « — 1 — 0a„ + {2(x + 1)(I + 1) + K^ + l)}a, +1 = 0.
(1838)
It can be shown by an argument similar to that used in Section
11a for the harmonic oscillator that for any values of X and I the
series whose coefficients are determined by this formula leads to
a function S(p) unacceptable as a wave function unless it breaks
off. For very large values of v the successive terms of an infinite
series given by 1838 approach the terms of the expansion
of e p , which accordingly represents the asymptotic behavior of
the series. This corresponds to an asymptotic behavior of
S(p) = e 2 p l L(p) similar to e 2 , leading to the infinity catastrophe
with increasing p.
Consequently the series must break off after a finite number
of terms. The condition that it break off after the term in p n ' is
seen from Equation 1838 to be
124 THE HYDROGEN ATOM [V18e
XZln'=0
or
X = n, where n = n' + I + 1. (1839)
n' is called the radial quantum number and n the fofoZ quantum
number. From its nature it is seen that n' can assume the values
0, 1, 2, 3, • • • . The values of n will be discussed in the next
section.
In this section we have found the allowed solutions of the
r equation to have the form R(r) = e 2 p l L(p), in which L(p) is
defined by the recursion formula 1838, with X = n. It will
be shown in Section 20 that these functions are certain associated
Laguerre functions, and a description of them will .be given in
Section 21.
18e. The Energy Levels. — Introducing for X its value as given
in Equation 1827, and solving for W y it is found that Equation
1839 leads to the energy expression
Wn =  W = _**£' =  Z 1 Wh> (1840)
h*n 2 n 2 n 2
in which
R = W£* and W H = Rhc.
h 3 c
This expression is identical with that of the old quantum theory
(Eq. 724), even to the inclusion of the reduced mass ju It is
seen that the energy of a hydrogenlike atom in the state repre
sented by the quantum numbers n', l } and m does not depend on
their individual values but only on the value of the total quantum
number n = n' + I + 1. Inasmuch as both n' and I by their
nature can assume the values 0, ,1, 2, • • • , we see that the
allowed values of n are 1, 2, 3, 4, • • • , as assumed in the old
quantum theory and verified by experiment (discussed in
Sec. 76).
Except for n = 1, each energy level is degenerate, being
represented by more than one independent solution of the wave
equation. If we introduce the quantum numbers n, l } and m
as subscripts (using n in preference to n') y the wave functions
we have found as acceptable solutions of the wave equation
may be written as
*nJm(r, 0, <p) = flni(r)ei m (tf)* w (*), (1841)
V19] LEGENDRE FUNCTIONS AND SURFACE HARMONICS 125
the functions themselves being those determined in Sections
186, 18c, and 18d. The wave functions corresponding to distinct
sets of values for n, Z, and m are independent. The allowed
values of these quantum numbers we have determined to be
m = 0, ±1, ±2, • • • ,
I = \m\, \m\ + 1, \m\ + 2, • • • ,
n = I + 1, I + 2, I + 3, • • • .
This we may rewrite as
total quantum number n — 1, 2, 3, • • • ,
azimuthal quantum number Z = 0, 1, 2, • • • , n — 1,
magnetic quantum number m = — Z, Z + 1, • • • ,—1, 0,
+ 1, • • • , +Z  1, +Z.
There are consequently 21 + 1 independent wave functions with
given values of n and Z, and n 2 independent wave functions with
a given value of n, that is, with the same energy value. The
2Z + 1 wave functions with the same n and Z are said to form a
completed subgroup, and the n 2 wave functions with the same n a
completed group. The wave functions will be described in the
following sections of this chapter.
A similar treatment applied to the wave equation with W
positive leads to the result that there exist acceptable solutions
for all positive values of the energy, as indicated by the general
discussion of Section 9c. It is a particularly pleasing feature of
the quantum mechanics that a unified treatment can be given
the continuous as well as the discrete spectrum of energy values.
Because of the rather complicated nature of the discussion of the
wave functions for the continuous spectrum (in particular their
orthogonality and normalization properties) and of their minor
importance for most chemical problems, we shall not treat them
further. 1
19. LEGENDRE FUNCTIONS AND SURFACE HARMONICS
The functions of & which we have obtained by solution of the
# equation are well known to mathematicians under the name of
associated Legendre functions. 2 The functions of d and <p are
1 See Sommerfeld, " Wave Mechanics," p. 290.
2 The functions of # for m =0 are called Legendre functions. The asso
ciated Legendre functions include the Legendre functions and additional
126 THE HYDROGEN ATOM [VlOa
called surface harmonics (or, in case cosine and sine functions of
<p are used instead of exponential functions, tesseral harmonics).
We could, of course, proceed to develop the properties of these
functions from the recursion formulas for the coefficients in the
polynomials obtained in the foregoing treatment. This would
be awkward and laborious, however; it is simpler for us to define
the functions anew by means of differential expressions or
generating functions, and to discuss their properties on this basis,
ultimately proving the identity of these functions with those
obtained earlier by application of the polynomial method.
19a. The Legendre Functions or Legendre Polynomials. — The
Legendre functions or Legendre polynomials Pj(cos #) = Pi(z)
may be defined by means of a generating function T(t, z) such
that
r( M)^X vl _' 2te + 1 , OH)
As in the case of the Hermite polynomials (Sec. lie), we
obtain relations among the polynomials and their derivatives by
differentiating the generating function with respect to t and to z.
Thus on differentiation with respect to t, we write
00
S"*""^
dt ^J ' (l2zt + t 2 )*
J~0
or
(1  2zt + t^lPit 1  1 ^ (z  t)%P t t l
i i
(the right side having been transformed with the use of Equation
191), and consequently, by equating coefficients of given powers
of t on the two sides, we obtain the , recursion formula for the
Legendre polynomials
(I + l)Pi+iC0 ~ (21 + l)zPi(z) + IPi^z) = 0. (192)
Similarly, by differentiation with respect to z, there is obtained
dz ~~ 2f l ~ (1 ~ 2zt + t 2 )*
i
functions (corresponding to I ml > 0) conveniently defined in terms of the
Legendre functions.
V19b] LEGENDRE FUNCTIONS AND SURFACE HARMONICS 127
or
(1 2zt + t*)2 f P' l t' = t2 f P l t',
i i
which gives the relation
PUz)  2zP[(z) + P'Uz)  P t (z) = (193)
involving the derivatives of the polynomials. Somewhat simpler
relations may be obtained by combining these. From 192
and 193, after differentiating the former, we find
zP[(z)  P'^z) ~ IPi(z) = (194)
and
P[ +l {z)  zP\(z)  (1 + l)P t (z) = 0. (195)
We can now easily find the differential equation which Pi{z)
satisfies. Reducing the subscript Z to Z — 1 in 195, and sub
tracting 194 after multiplication by z, we obtain
(i  * 2 )p; + up 1  zp,_! = o,
which on differentiation becomes
Tz{ (l ~ z2)d ^r\ + lPl{z) + lzP ' i{z) ~ lp ' i{z) = °
The terms in P\ and P[_i may be replaced by l 2 P t , from 194,
and there then results the differential equation for the Legendre
polynomials
~{(1 ~ * 2 )~^} + 1(1 + DftW = 0. (196)
19b. The Associated Legendre Functions. — We define the
associated Legendre functions of degree Z and order \m\ (with
values I = 0, 1, 2, • • • and \m\ = 0, 1, 2, • • • , Z) in terms of
the Legendre polynomials by means of the equation
PlT'CO  (1  z 2 )' ml/2 S p '(z). (197)
[It is to be noted that the order \m\ is restricted to positive values
(and zero) ; we are using the rather clumsy symbol \m\ to represent
the order of the associated Legendre function so that we may
later identify m with the magnetic quantum number previously
128 THE HYDROGEN ATOM [V19b
introduced.] The differential equation satisfied by these func
tions may be found in the following way. On differentiating
Equation 196 \m\ times, there results
rf\m\+2p ( z \ d^ m ^ l Pi(z\
(i  * 2 )^p£p " 2(M + i)^^£P
+ {l(l + l) m(H + DI^P = (198)
d lml Pi(z)
as the differential equation satisfied by — m , • With the
use of Equation 197 this equation is easily transformed into
_ *£%p _ »£«■> + {«, + „  J±Jjpro)
= 0, (199)
which is the differential equation satisfied by the associated
Legendre function P\ ml (z).
This result enables us to identify 1 the & functions of Section
18c (except for constant factors) with the associated Legendre
functions, inasmuch as Equation 199 is identical with Equation
1819, except that P(z) is replaced by P\ ml (z) and fi is replaced
by 1(1 + 1), which was found in Section 18c to represent the
characteristic values of £. Hence the wave functions in #
corresponding to given values of the azimuthal quantum number I
and the magnetic quantum number m are the associated Legendre
functions P[ ml (z).
The associated Legendre functions are most easily tabulated by
the use of the recursion formula 192 and the definition J 97,
together with the value Pq{z) = 1 as the starting point. A
detailed discussion of the functions is given in Section 21.
For some purposes the generating 'function for the associated
Legendre functions is useful. It is found from that for the
Legendre polynomials to be
T, ,(z t) = ^?P™(z)t< = (ZMWl**) 1  1 ^ 1 " 1
J«i^» l > ~ ^n W*  2i»i(m)!(l  2zt + «*)i"+M
iM
(1910)
1 The identification is completed by the fact that both functions are formed
from polynomials of the same degree.
V20a] THE LAGUERRE POLYNOMIALS 129
In Appendix VI it is shown that
(1911)
c +i (ofoir ?*i,
FTKzWWdz = { 2 (l + \m\)\ _
Using this result, we obtain the constant necessary to normalize
the part of the wave function which depends on &. The final
form for 0(#) is
e W = ^?L^> <^M>i p , W) . (19 _ 12)
Problem 191. Prove that the definition of the Legendre polynomials
Poto  1, )
""■"ar^ '>•*••■ I (,M "
is equivalent to that of Equation 191.
Problem 192. Derive the following relations involving the associated
Legendre functions:
(i  *' W'(*) = ^jfjM+'iM  ^pTj pl "i(*), (19 ~ 14 >
(i  «w« w  « + H^ (l + + W + % ) 
aH^H^)^, (19 _ 15)
and
^' (Z) (2? + 1) Pl  l{2) + (21 + 1) Pl+M ' (1 ^ 16)
20. THE LAGUERRE POLYNOMIALS AND ASSOCIATED LAGUERRE
FUNCTIONS
20a. The Laguerre Polynomials. — The Laguerre polynomials
of a variable p, within the limits ^ p ^ oo , may be defined by
means of the generating function
^«)2^ r r^ (2(M)
r=0
To find the differential equation satisfied by these polynomials
Lr(p), we follow the now familiar procedure of differentiating the
130 THE HYDROGEN ATOM [V20a
dU
generating function with respect to u and to p. From — we
obtain
2 Lr(o) t = e iy _P_ _ P" , _J_\
(r  1) ! 1  u\ 1  « (1  w) 2 ^ 1  u/
or
r r
from which there results the recursion formula
L r+l (p) + ( p  1  2r)L r ( P ) + r 2 L r _!( P ) = 0. (202)
Similarly from — we have
^i r! 1 — Uj^j r\
r r
or
L' r (p)  ^r'i(p) + rLri(p) = 0, (203)
in which the prime denotes the derivative with respect to p.
Equation 203 may be rewritten and differentiated, giving
KM =(r + l){L;(p) L r (p)\
and
L' r ' +l (p) =(r + l){L;'(p)L;(p)l,
with similar equations for L' r+2 (p) and L" +2 (p). Replacing r by
r + 1 in Equation 202 and differentiating twice, we obtain
the equation
KM + (P ~ 3  2r)L£ 1 (p) + (r + l) 2 L^(p) + 2L r ' +1 (p) = 0.
With the aid of the foregoing expressions this is then transformed
into an equation in L r (p) alone,
pL'/ip) + (1  p)L' r {p) + rL r (p) = 0, (204)
which is the differential equation for the rth Laguerre polynomial.
d r
Problem 201. Show that L f (p) = e p —(p r e" p ).
dp r
VSM)b] THE LAGUERRE POLYNOMIALS 131
20b. The Associated Laguerre Polynomials and Functions.—
The sth derivative of the rth Laguerre polynomial is called the
associated Laguerre polynomial of degree r — s and order s:
L' r ( P ) = £Mp) (206)
The differential equation satisfied by L' r (p) is found by differ
entiating Equation 204 to be
pL' r "{ P ) + (s + 1  p)L' r \p) + (r  s)L< r {p) = 0. (206)
If we now replace r by n + I and s by 21 + 1, Equation 206
becomes
pL^V'Xp) + {2(J + 1)  p)L^\p)
+ (n  i  l)L 2 n VV(p) = 0. (207)
On comparing this with Equation 1837 obtained in the treat
ment of the r equation for the hydrogen atom by the polynomial
method, we see that the two equations are identical when
Ln+i l (p) is identified with L{p) and the parameter X is replaced
by its characteristic value n. The polynomials obtained in the
solution of the r equation for the hydrogen atom are hence the
associated Laguerre polynomials of degree n — I — 1 and of
order 21 + 1. Moreover, the wave functions in r are, except for
normalizing factors, the functions
These functions are called the associated Laguerre functions.
We shall discuss them in detail in succeeding sections.
It is easily shown from Equation 201 that the generating
function for the associated Laguerre polynomials of order s is 1
00 pU
u.(p, «)  2^r ur  ( " V' iiuy^ ' (20_8)
The polynomials can also be expressed explicitly:
nll
L 2i +Hp) = ^ /_ n *+i i(n + l)\}>
L n+l ( P ) j^k i; (n _ j _ ! _ fc)!(2 j + x + fc)!fcI P •
fc=0
(209)
1 This was given by Schrodinger in his third paper, Ann. d. Phys. 80, 486
(1926).
132 THE HYDROGEN ATOM [V21a
In Appendix VII, it is shown that the normalization integral
for the associated Laguerre function has the value
Vp« {L#i>(p) } Vdp = ^iY !)! ' (2 ° 10)
1
the factor p 2 arising from the volume element in polar coordinates.
From this it follows that the normalized radial factor of the wave
function for the hydrogen atom is
with
*"« = V(S) 3 2:i(n + 0!}' e ^ P ' L " +V(p) ' (2 ° U)
p  2ar = **&  M,. (20 12)
r n/i 2 na
Problem 202. Derive relations for the associated Laguerre polynomials
and functions corresponding to those of Equations 202 and 203.
21. THE WAVE FUNCTIONS FOR THE HYDROGEN ATOM
21a. Hydrogenlike Wave Functions. — We have now found
the wave functions for the discrete stationary states of a one
electron or hydrogenlike atom. They are
+nim(r, tf, tp) = RnMGUmmM, (211)
with
and
*mM = )«**, (212)
^{^^^w^r^ ^ (2i  3)
Rn,{r) = [w &^
Z 1)!
+ 0MJ
in which
VVLJSiKp), (214)
p = — r (215)
na
and
«o =
47T 2 /X6 2
a being the quantity interpreted in the old quantum theory a3
the radius of the smallest orbit in the hydrogen atom. The
V21a] THE WAVE FUNCTIONS FOR THE HYDROGEN ATOM 133
functions P[ mI (cos #) are the associated Legendre functions
discussed in Section 19, and the functions L^f^p) are the asso
ciated Laguerre polynomials of Section 20. The minus sign in
Equation 214 is introduced for convenience to make the function
positive for small values of r.
The wave functions as written here are normalized, so that
fo" fj fo"** 1 ^' *' *)*»'"( r ' *' ^ sin MrfMr = 1  ( 21 ~ 6 )
Moreover, the functions in r, #, and <p are separately normalized
to unity:
£'*X<p)*„Md<p = 1,
f r {Qim(#)} 2 sin #d# = 1,1 (217)
f °° {R nl (r)} 2 r 2 dr = 1.
They are also mutually orthogonal, the integral
J["J[* f*"+nim(r, &, <p)+ n rAr, *, <f)r 2 sin *d<pd*dr
vanishing except for n = n', I = l' } and m = m' '; inasmuch as if
m 7* m\ the integral in <p vanishes; if m = m! ', but I ^ V ', the
integral in # vanishes; and if m = m' and I = I' ', but n ^ n',
the integral in r vanishes.
Expressions for the normalized wave functions for all sets of
quantum numbers out to n — 6, I = 5 are given in Tables 211,
212, and 213.
The functions $m(<p) are given in both the complex and the
real form, either set being satisfactory. (For some purposes
one is more convenient, for others the other.)
Table 211. — The Functions & m M
<t> (^) = — y~ Or $oO) = — J=^
V27T V 2ir
$i(v?) = — p=c»* or $icos(^) = —^ cos <p
1 1
*iW = — y=£r tip or *tain(*0 = —7= sin ip
\/2w V*
134 THE HYDROGEN ATOM [V21a
Table 211. — The Functions *m(¥>). — (.Continued)
* 2 (^)  ~" pzze™ Or *2coe(¥>)  — 7= cos 2<p
\Z2tt y/ic
*_ 2 (*>) = — —e~ ivt> or **in(*0 = — 7^ sin 2<p
y/2v y/v
Etc.
Table 212.— The Wave Functions e* m (0)
(The associated Legendre functions normalized to unity)
I = 0, s orbitals:
e o(t» = y
/ » 1, p orbitals:
©io(<>) = r cos t>
V3
e lafcl (0) = —  sin
I = 2, d orbitals:
a/10
G 2O (0)  ~ — (3 cos 2 1?  1)
4
/15 .
62*1 (#) = • sin cos c q
a/15
e 2 . 2 (0) = —— sin 2
4
1 = 3,/ orbitals :
, s 3Vl4/5 , \
e 8O (0) = v (  cos 3  cos 1
v/42
B,*i(t?) = ^— sin 0(5 cos 2 01)
o
O,* 2 (0) = ~ — sin 2 cos
a/70
G 8 *8 W = ~~ sin 3
8
J = 4, gr orbitals:
9a/2 /35 \
e 4O (0) = ~ ( ■— cos 4 010 cos 2 Hi]
, N 9\/l0 (l % \
9i*i W = —  — sin I  cos 3 — coft I
V21a] THE WA VE F UNCTIONS FOR THE H YDROGEN A TOM 135
Table 212. — The Wave Functions Gj m (#).— (Continued)
3\/5
~ /on V gi n 2 ,y(7 CO g2 ^ _ 1)
t>4*
■2W
= g on,
n<
e 4 
>(*)
3^70 . ,
8
'*
e 4 
.4(1?)
3\/35 . A
t>
cost?
I = 5, h orbitals:
, x 15\/22/21 14 t \
©6o(#) = — ~ — I ™ cos 6 1?  — cos 8 1? f cos 1? )
16 \5
6ui(« =
'165
16
 sin #(21 cos 4 #  14 cos 2 tf + 1)
'1155
8
/770
32
sin 2 1?(3 cos 3 1> — cos #)
sin 3 # (9 cos 2  1)
x 3V385 . ,
0»*4W = — — — sin 4 & cos #
16
, N 3^154 . K
e M W = 32 sin 6 *
Table 213. — The Hydrogenlike Radial Wave Functions
n = 1, K shell:
— £
I = 0,1s ffio(r) = (Z/a )K'2e 2
n = 2, L shell:
(Z/o )»
Z = 0, 2s ft, (r) =
I = 1, 2p fl„(r) =
n = 3, M shell:
J = 0, 3s /2» (r) =
J  1, 3p «,i(r) 
2\/2
(Z/a )M
2y/E
(Z/oo)»
9\/3
(Z/ao)*
9V6
(2  P )e 2
pe
(6  6 P + P 2 )e
(4  P ) P e 2
,^« 2
J = 2, 3d * M (r)  ^£>>e~ 2
9V30
136 THE HYDROGEN ATOM [V21a
Table 213. — The Hydrogenlike Radial Wave Functions. — (Continued)
n = 4, N shell:
= 0, 4 8 R A0 (r) = (Z/ ^ o)/ (24  36p + 12 P 2  p 3 )e~ 2
= 1, 4p R Al (r) = ^L(20  10p + P *)pe 2
32 VI 5
= 2, 4d /2„(r) = {Z/ao) _ (6  p^e" 2
96V5
= 3,4/ ft.fr) ^V*
= 5, O^hell:
(Z la )M £
= 0, 5s R 60 (r) =  ' " (120 ~ 24 °p + 12 °p 2 ~ 2 °P 3 + P 4 )* 2
300 V5
(7 In )^ — 
= 1, 5p « 6 i(r) = ^==.(120  90 P + 18 P 2  P 3 ) P e 2
150 V 30
/w/ yi^ __ p
= 2, bd R M (r) = ^— °;1=(42  14 P + P 2 ) P 2 e 2
150 V 70
(7 In \¥* — 
= 3, 5/ ft 63 (r) = ^^(8  P ) P 3 e 2
300 V 70
= 4, 5gr R b i(r) = 7=P 4 e z
900V70
= 6, P shell:
(Z/a ) H
■ 0, 6s fleo(r) = ^ (720  1800 P + 1200p 2  300p 3 + 30 P 4
2160V6
6V 2
P 6 )e
= 1, 6p R t i(jr) = 7= (840  840 P + 252 P 2  28 P 3 + P 4 ) P e 2
432V 2 10
= 2, 6d fl 62 (r) = °/_ (336  168 P + 24 P 2  P 3 ) P 2 e~ 2
864V105
= 3, 6/ R n (r) = (Z/a ° ) / L(72  18 P + P 2 )p 3 e~2
2592 V 35
= 4, 60 R*(r) = (Z/ao) ^ (10  p^e"^
12960V7
= 5, 6/i /?66(r) = 7==P 6 e z
12960 V77
V21a] THE WA VE FUNCTIONS FOR THE HYDROGEN ATOM 137
The wave functions 9j m (#) given in Table 212 are the asso
ciated Legendre functions P m (cos #) normalized to unity. The
functions P[ m ^ (cos &) as usually written and as defined by
Equations 191 and 197 consist of the term sin lml # and the
polynomial in cos # multiplied by the factor
(* + M)i or q + M + p!
Jl + \m\\Jl  \m\\ Jl + \m\ + lV/ (j  \m\  l V
as m + I is even or odd. Expressions for additional associated
Legendre functions are given in many books, as, for example,
by Byerly. 1 Numerical tables for the Legendre polynomials
are given by Byerly and by Jahnke and Emde. 2
Following Mulliken, we shall occasionally refer to oneelectron
orbital wave functions such as the hydrogenlike wave functions
of this chapter as orbitals. In accordance with spectroscopic
practice, we shall also use the symbols s, p y d, /, g, • • • to
refer to states characterized by the values 0, 1, 2, 3, 4, • • • ,
respectively, of the azimuthal quantum number l y speaking, for
example, of an s orbital to mean an orbital with 1 = 0.
In the table of hydrogenlike radial wave functions the poly
nomial contained in parentheses represents for each function
the associated Laguerre polynomial Ll^ip), as defined by
Equations 201 and 205, except for the factor
(n + i)!/(ni 1)!,
which has been combined with the normalizing factor and
reduced to the simplest form. It is to be borne in mind that
the variable p is related to r in different ways for different
values of n.
The complete wave functions 4<nim(r, #, <p) for the first three
shells are given in Table 214. Here for convenience the variable
p = 2Zr/na has been replaced by the new variable <r, such that
n Z
W. E. Byerly, "Fourier's Series and Spherical Harmonics," pp. 151,
159, 198, Ginnand Company, Boston, 1893.
2 W. E. Byerly, ibid., pp. 278281; Jahnke and Emde, "Funktionen
tafeln," B. G. Teubner, Leipzig, 1933.
138 THE HYDROGEN ATOM [V21a
The relation between <r and r is the same for all values of th°
quantum numbers. The real form of the <p functions is used.
The symbols p x , p v , p x , d x+v , d y + g , d x + x , d xyi and d z are introduced
for convenience. It is easily shown that the functions ^ np ,,
\l/ nPy , and \l/ nPa are identical except for orientation in space, the
three being equivalently related to the x, y, and z axes, respec
tively. Similarly the four functions tfw, +1 ,, ^nd v+tf tnd x+t , and
$nd tv are identical except for orientation. The fifth d function
\l/ ndt is different.
Table 214. — Hydrogenlike Wave Functions
K Shell
n = 1, I = 0, w = 0:
= — (Y *
\Ztt \ a oJ
//Shell
n = 2, I = 0, m = 0:
1 /Z\H 
+u = 7=() (2<r)6 2
4 V2* \ a o/
0:
*i p = 7= (  ) ° e cos i>
4V2* \«o/
= ±1:
i /zV*
^2 Pj = 7= I — J <r
4 V 2tt \Oo/
l /zV* s .
^ 2p = = ( — J ae 2 sin i? sin v
4V2* \ a °/
n = 2, J = 1, m = 0:
n = 2, / = 1, m = ±1:
i /zV __^
oe 2 sin # cos <p
M Shell
n = 3, / = 0, m = 0:
n  3, / = 1, m = 0:
i /zV 
*i. = 7= f  ) (27  18<r + 2<r 2 )e 3
~ /3t V° /
81 V^
n = 3, J = 1, m = ±1:
V2
^sp, = 7= ( — ) (6 — a)ae 3 COS t?
(6  <r)ae 3 I
V?" /Z\
^i p , = 7= ( — 1 (6 — <r)<re 6 sin t> cos ^
V21b] THE WAVE FUNCTIONS FOR THE HYDROGEN ATOM 139
Table 214. — Hydrogenlike Wave Functions. — (Continued)
^ / Z \H £
tzp y = 7=[ — ) (6 — <r)ae 6 sin # sin <p
81 VV \«o/
n =: 3 ? / = 2, m = 0:
81 V^ V (
^is x+t = p I — ] (r 2 e 3 sin #cos t? cos v»
^sd v+# = 7= I —  * 2 e sin # cos # sin <p
1 /ZV* 
*, d , = =, [  ) <r 2 e 3 (3 cos 2 tf  1)
81V6ir \°o/
n = 3, J = 2, w = ± 1 :
— [ — 1 <r 2 e * sin # cos t?
81 Vt \ a °/
8lV^\ a °/
n = 3, J = 2. m = ±2:
1 /ZV*  5
^3d Itf = 7=  — I <r l e 3 sin 2 # cos 2<p
81V2x \ a V
i /zV* 5
^sd, +tf = 7=r [ —  <r 2 e 3 sin 2 1? sin %p
81 V2ir \ fl o/
Z
with <r = — r.
a
21b. The Normal State of the Hydrogen Atom. — The proper
ties of the hydrogen atom in its normal state (Is, with n = 1,
I = o, m = 0) are determined by the wave function
1 —
^ 100 = : —  e ao .
'jra:
The physical interpretation postulated for the wave function
1 ^
requires that \^*^ = — ge °° be a probability distribution function
TTCLq
for the electron relative to the nucleus. Since this expression
is independent of & and <p f the normal hydrogen atom is spheri
cally symmetrical. The chance that the electron be in the
1 £
volume element r 2 dr sin &d&d(p is — z e °*r 2 dr sin &d$d<p, which
TTCLq
is seen to be independent of # and <p for a given size of the volume
element. This spherical symmetry is a property not possessed
by the normal Bohr atom, for the Bohr orbit was restricted to a
single plane.
140
THE HYDROGEN ATOM
[V21b
By integrating over # and <p (over the surface of a sphere),
we obtain the expression
4 ^
D(r)dr = ~^r 2 e ao dr
cl
as the probability that the electron lie between the distances
r and r + dr from the nucleus. The radial distribution function
2r
100
D\Qo(r) = —jr 2 e ao is shown in Figure 211 (together with ^
a
and ^ioo) as a function of r, the distance from the nucleus. It
* i
\
..,
k
4jrrWf
f 1 ^
1 1 1 1 1 1
(
) 1.0 2.0 3.0%
Fig. 211. — The functions \f/, \p*\p, and 4x7 VV for the normal hydrogen
atom. The dashed curve represents the probability distribution function for a
Bohr orbit.
is seen that the probability that the electron remain within about
1 A of the nucleus is large ; that is, the "size" of the hydrogen atom
is about the same as given by the Bohr theory. Indeed, there is
a close relation; the most probable distance of the electron from
the nucleus, which is the value of r at which D(r) has its maximum
value, is seen from Figure 211 to be a = 0.529A, which is just
the radius of the normal Bohr orbit for hydrogen.
V21b] THE WA VE FUNCTIONS FOR THE HYDROGEN ATOM 141
The distribution function itself is not at all similar to that
for a circular Bohr orbit of radius a , which would be zero every
where except at the point r = a . The function ^ 00 has its
maximum value at r = 0, showing that the most probable
position for the electron is in the immediate neighborhood of
the nucleus; that is, the chance that the electron lie in a small
volume element very near the nucleus is larger than the chance
that it lie in a volume element of the same size at a greater
distance from the nucleus. 1 It may be pointed out that a Bohr
orbit in the form of a degenerate line ellipse, obtained by giving
the azimuthal quantum number k of the old quantum theory
the value instead of the value 1, leads to a distribution function
resembling the wavemechanical one a little more closely. This
is shown in Figure 211 by the dashed curve. The average
distance of the electron from the nucleus, given by the equation
fnlm = Ji ' ttnlnrtnl m r*dr SU1 M&d<p y (21~8)
is found in this case to be equal to ^a . This is also the value
calculated for the Bohr orbit with k — 0; in fact, it will be shown
in the next section that for any stationary state of the hydrogen
atom the average value of r as given by the quantum mechanics
is the same as for the Bohr orbit with the same value of n and
with k 2 equal to 1(1 + 1). It will also be shown in Chapter XV
that the normal hydrogen atom has no orbital angular momen
tum. This corresponds to a Bohr orbit with fc = but not with
k = 1. The rootmeansquare linear momentum of the electron
is shown in the next section to have the value 2irixe 2 /h y which is
the same as for the Bohr orbit. We may accordingly form a
rough picture of the normal hydrogen atom as consisting of an
electron moving about a nucleus in somewhat the way cor
responding to the Bohr orbit with n = 1, k = 0, the motion
being essentially radial (with no angular momentum), the
amplitude of the motion being sufficiently variable to give rise
to a radial distribution function D(r) extending to infinity,
though falling off rapidly with increasing r outside of a radius
of 1 or 2A, the speed of the electron being about the same as in
the lowest Bohr orbit, and the orientation of the orbit being
1 The difference between the statement of the preceding paragraph and
this statement is the result of the increase in size of the volume element
4rr 2 dr for the former case with increasing r.
142
THE HYDROGEN ATOM
[V21c
sufficiently variable to make the atom spherically symmetrical.
Great significance should not be attached to such a description.
We shall, however, make continued use of the comparison of
wavemechanical calculations for the hydrogen atom with
the corresponding calculations for Bohr orbits for the sake of
convenience.
5
t
4
\
3
A
2
i \
1
\ \^^
n1, 10
I
\\
n2, 10
?
\ \
A x ~
\
n3, 10
1
\^
n2,ll
1
n3, 11
. L 1 1
1..  J 1 L L I. 
8 10 12 14 16 18 20
?•+■
Fig. 212. — Hydrogenatom radial wave functions R n i(r) for n = 1, 2, and 3 and
I = and 1.
21c. Discussion of the Hydrogenlike Radial Wave Functions.
The radial wave functions R n i(r) for n = 1, 2, and 3 and I =
and 1 are shown plotted in Figure 212. The abscissas represent
values of p; hence the horizontal scale should be increased by the
factor n in order to show R (r) as functions of the electronnucleus
distance r. It will be noticed that only for s states (with I = 0)
is the wave function different from zero at r = 0. The wave
function crosses the p axis n — I — 1 times in the region between
p = and p = oo .
V21c] THE WA VE FUNCTIONS FOR THE HYDROGEN A TOM 143
The radial distribution function
Dni(r)=rMftni(0} 2 (219)
is represented as a function of p for the same states in Figure
213. It is seen from Figures 212 and 213 that the probability
distribution function ^V> which is spherically symmetrical
for s states, falls off for these states from a maximum value at
r = 0. We might say that over a period of time the electron
08
04
08
I 04
E o
c
Gc
t^ 04
^
08
04
04
n1, 10
10
9"
12 14 16 18 20
Fig. 213. — Electron distribution functions 4xr t [R n i(r)] 1 for the hydrogen atom.
may be considered in a hydrogen atom in the normal state to
form a ball about the nucleus, in the 2s state to form a ball and
an outer shell, in the 3s state to form a ball and two concentric
shells, etc. The region within which the radial distribution
function differs largely from zero is included between the values
of r at perihelion and aphelion for the Bohr orbit with the same
value of n and with k 2 = 1(1 + 1), as is shown by the heavy
horizontal line for each curve in Figure 213, drawn between the
minimum and maximum values of the electronnucleus distance
144 THE HYDROGEN ATOM [V21c
for this Bohr orbit in each case. For these s orbits (with k = 0)
the heavy line extends to r = 0, corresponding to a line ellipse
with vanishingly small minor axis, in agreement with the large
value of yf/*}// at r = 0. For states with I > 0, on the other hand,
\f/*\l/ vanishes at r = 0, and similarly the minimum value of r
for the Bohr orbits with k = \/l(l + 1) is greater than zero.
The average distance of the electron from the nucleus, as given
by Equation 218, is found on evaluating the integral to be
Tnlm — 7*
M{'^}] <«»»
The corresponding values of p are represented by vertical lines in
Figure 213. From this expression it is seen that the size of the
atom increases about as the square of the principal quantum
number n, f n im being in fact proportional to n 2 for the states
with 1 = and showing only small deviations from this propor
tionality for other states. This variation of size of orbit with
quantum number is similar to that of the old quantum theory,
the timeaverage electronnucleus distance for a Bohr orbit
being
rrao
{ i +&$)}■ < 2i  u >
which becomes identical with the wavemechanical expression
if k 2 is replaced by 1(1 + 1), as we have assumed in the foregoing
discussion.
Formulas for average values of various powers of r are given
below. 1 It is seen that the wavemechanical expressions as a
rule differ somewhat from those of the old quantum theory,
even when k 2 is replaced by 1(1 + 1).
Average Values* of r 9
Wave Mechanics
ft'+if 1 *^}]
* Expressions for r are given in Equations 2110 and 2111.
1 1. Waller, Z. f. Phys. 38, 635 (1926); expressions for
are given by J. H. Van Vleck, Proc. Roy. Soc. A 143, 679 (1934).
(?)(*)
V21c] THE WAVE FUNCTIONS FOR THE HYDROGEN A TOM 145
Average Values of r*. — (Continued)
Z
a Q n 2
(A
\V aWd + H
Z 3
3 W + HW + 1)
2 ( 3n 2 j
Old Quantum Theory
¥{■+££)}
Z
aon 2
M —
\r 3 / ajn 3 /c 3
\r 4 / a,!n 3 /c 6
To illustrate the use of these formulas, let us calculate the
average potential energy of the electron in the field of the
nucleus. It is
Vmm =111 +: im ~+«i m r>dr sin #d#d<p
= Ze
Z 2 e 2
(2112)
Now the total energy W , which is the sum of the average ki
netic energy f and the average potential energy V, is equal to
— Z 2 e 2 /2aon 2 . Hence we have shown that the total energy is
just onehalf of the average potential energy, and that the average
146 THE HYDROGEN ATOM [V21d
kinetic energy is equal to the total energy with the sign changed,
i.e.,
'•'  !£■ (21  13)
This relation connecting the average potential energy, the
average kinetic energy, and the total energy for a system of
particles with Coulomb interaction holds also in classical mechan
ics, being there known as the virial theorem (Sec. 7a).
Now we may represent the kinetic energy as
T = ±(vl + Vl + Vl),
in which p X) p V) and p z represent components of linear momentum
of the electron and nucleus relative to the center of mass (that
is, the components of linear momentum of the electron alone
if the small motion of the nucleus be neglected). Hence the
average value of the square of the total linear momentum
P 2 = vl + 7>l + Vl is equal to 2/x times the average value of the
kinetic energy, which is itself given by Equation 2113 for both
wave mechanics and old quantum theory. We thus obtain
**» = "2o^ = \iar) (21 " 14)
as the equation representing the average squared linear momen
tum for a hydrogenlike atom in the wave mechanics as well as in
the old quantum theory. This corresponds to a rootmean
square speed of the electron of
/= 2*Ze 2
nh
(2115)
which for the normal hydrogen atom has the value 2.185 X 10 8
cm/sec.
Problem 211. Using recursion formulas similar to Equation 202 (or
in some other way) derive the expression for f«j m .
21d, Discussion of the Dependence of the Wave Functions on
the Angles d and $. — In discussing the angular dependence of
hydrogenlike wave functions, we shall first choose the complex
form of the functions <£>(^>) rather than the real form. It will be
shown in Chapter XV that there is a close analogy between the
V21d] THE WA VE FUNCTIONS FOR THE HYDROGEN A TOM 147
stationary states represented by these wave functions and the
Bohr orbits of the old quantum theory in regard to the orbital
angular momentum of the electron about the nucleus. The
square of the total angular momentum for a given value of I
h 2
is 1(1 + 1)t2> an( * the component of angular momentum along
the z axis is mh/2Tr, whereas the corresponding values for a Bohr
orbit with quantum numbers nkm are k 2 h 2 /4w 2 and mh/2w,
respectively. We interpret the wave functions with a given
value of I and different values of m as representing states in which
the total angular momentum is the same, but with different
orientations in space.
It can be shown by a simple extension of the wave equation
to include electromagnetic phenomena (a subject which will
not be discussed in this book) that the magnetic moment asso
ciated with the orbital motion of an electron is obtained from the
orbital angular momentum by multiplication by the factor
e/2m c, just as in the classical and old quantum theory (Sec. Id).
The component of orbital magnetic moment along the z axis is
hence mj ; and the energy of magnetic interaction of this
moment with a magnetic field of strength H parallel to the z axis
he rj
is m. H.
4rt"moC
In the old quantum theory this spatial quantization was sup
posed to determine the plane of the orbit relative to the fixed
direction of the z axis, the plane being normal to the z axis for
m = ±k and inclined at various angles for other values of m.
We may interpret the probability distribution function ^*^ in a
similar manner. For example, in the states with m = ±1
the component of angular momentum along the z axis, mh/2w,
is nearly equal to the total angular momentum, \/l(l + l)h/2w,
so that, by analogy with the Bohr orbit whose plane would be
nearly normal to the z axis, we expect the probability distribution
function to be large at & = 90° and small at & = 0° and 180°.
This is found to be the case, as is shown in Figure 214, in which
there is represented the function {Qim(&)} 2 for m = ±1 and for
I = 0, 1, 2, 3, 4, and 5. It is seen that as I increases the prob
ability distribution function becomes more and more concen
trated about the xy plane.
148
THE HYDROGEN ATOM
[V21d
Fig. 214.— Polar graphs of the function [Oi m W] 2 for m = ± I and I = 0,
1, 2, 3, 4, and 5, showing the concentration of the function about the xy plana
with increasing I.
V21d] THE WA VE FUNCTIONS FOR THE HYDROGEN A TOM 149
The behavior of the distribution function for other values of m
is similarly shown in Figure 215, representing the same function
f or I = 3 and m = 0, ± 1, ±2, ±3. It is seen that the function
tends to be concentrated in directions corresponding to the
plane of the oriented Bohr orbit (this plane being determined
only to the extent that its angle with the z axis is fixed).
With the complex form of the <p functions, these figures
represent completely the angular dependence of the probability
Fia. 215.
Polar graphs of the function [6{ OT (t?)] 2 for I
±2, and ±3.
3 and m = 0, ±1.
distribution function, which is independent of <p. The alterna
tive sine and cosine functions of <p correspond to probability
distribution functions dependent on <p in the way corresponding
to the functions sin 2 m<p and cos 2 nap. The angular dependence
of the probability distribution function for s and p orbitals in
the real form (as given in Table 214) is illustrated in Figure 216.
It is seen that, as mentioned before, the function s is spherically
symmetric, and the functions p x , p V) and p s are equivalent except
for orientation. The conditions determining the choice of wave
functions representing degenerate states of a system will be
discussed in the following chapter.
150
THE HYDROGEN ATOM
[V21d
A useful theorem, due to Unsold, 1 states that the sum of the
probability distribution functions for a given value of I and all
values of m is a constant; that is,
+i
X Oim(#)*m*(<p)e lm (&)* m (<p) = constant. (2116)
Px Py
Fig. 216. — Polar representation of the absolute values of the angular wave
functions for a and p orbitals. The squares of these are the probability distribu
tion functions.
The significance of this will be discussed in the chapter dealing
with manyelectron atoms (Chap. IX).
Problem 212. Prove Uns61d's theorem (Eq. 2116).
1 A. Unsold, Ann. d. Phys. 82, 355 (1927).
CHAPTER VI
PERTURBATION THEORY 1
In case that the wave equation for a system of interest can be
treated by the methods described in the preceding chapters, or
can be rigorously treated by any amplification of these methods,
a complete wave mechanical discussion of the system can be
given. Very often, however, such a procedure cannot be carried
out, the wave equation being of such a nature as to resist accurate
solution. Thus even the simplest manyelectron systems, the
helium atom and the hydrogen molecule, lead to wave equations
which have not been rigorously solved. In order to permit
the discussion of these systems, which more often than not are
those involved in a physical or especially a chemical problem,
various methods of approximate solution of the wave equation
have been devised, leading to the more or less accurate approxi
mate evaluation of energy values and wave functions. Of these
methods the first and in many respects the most interesting is
the beautiful and simple wavemechanical perturbation theory,
developed by Schrodinger in his third paper in the spring of 1926.
It is especially fortunate that this theory is very much easier
to handle than the perturbation theory which is necessary for
the treatment of general problems in classical dynamics.
Before we can discuss this method, however, we need certain
mathematical results concerning the possibility of expanding
arbitrary functions in infinite series of normalized orthogonal
functions. These results, which are of great generality and
widespread utility, we shall discuss in the next section without
attempting any complete proof.
22. EXPANSIONS IN SERIES OF ORTHOGONAL FUNCTIONS
The use of power series to represent certain types of functions
is discussed in elementary courses in mathematics, and the
theorems which state under what conditions the infinite series
1 A generalized perturbation theory will be discussed in Section 27a.
151
152 PERTURBATION THEORY [VI22
obtained by formal methods converge to the functions they are
meant to represent are also well known. An almost equally
useful type of infinite series, which we shall use very frequently,
is a series the terms of which are members of a set of normalized
orthogonal functions each multiplied by a constant coefficient.
If fo(x) t fi(x), fi{x) y • • • are members of such a set of normal
ized orthogonal functions, we might write as the series
(p(x) = aafo(x) + aifi(x) + ^2/2(2) + • • *
= Xanf n (z). (221)
n=0
If the series converges and has a definite sum <p(x), we may express
Equation 221 by saying that the infinite series on the right of
the equation represents the function <p(x) in a certain region of
values of x. We may ask if it is possible to find the coefficients
a n for the series which represents any given function <p(x). A
very simple formal answer may be given to this question. If
we multiply both sides of Equation 221 by ft(x) and then
integrate, assuming that the series is properly convergent so
that the termbyterm integration of the series is justified,
then we obtain the result
£*(x)ft{x)dx = Oft, (222)
since
£mx)f n (x)dx = if n * *, ) (22 _ 3)
= 1 if n = k.)
a ^ x ^ b defines the orthogonality interval for the functions
/(*)■
In many cases the assumptions involved in carrying out
this formal process are not justified, since the series obtained may
either not converge at all or converge to a function other than
v?(x). Mathematicians have studied in great detail the condi
tions under which such series converge and have proved
theorems which enable one to make a decision in all ordinary
cases. For our purposes, however, we need only know that such
theorems exist and may be used to justify all the expansions
which occur in this and later chapters.
VI22] EXPANSIONS IN SERIES OF ORTHOGONAL FUNCTIONS 153
The familiar Fourier series is only one special form of an
expansion in terms of orthogonal functions. Figure 221,
which gives a plot of the function
<p(x) = 1 for < x < t, ) (22rA)
<p(x) = 1 for t < x < 2tt,/
together with the first, third, and fifth approximations of its
Fourierseries expansion
<p(x) = a + «i sin x + bi cos x + a 2 sin 2x +
62 cos 2x + • • • , (225)
illustrates that a series of orthogonal functions may represent
even a discontinuous function except at the point of discontinuity.
Fig. 221. — The function <p{x) = + 1 f or < x < w, 1 for tt < x < 2ir, and
the first, third, and fifth Fourierseries approximations to it, involving terms to
oin x, sin 3x, and sin 5x, respectively.
If we had evaluated more and more terms of Equation 225,
the series would have approached more and more closely to the
function <p(x), except in the neighborhood of the discontinuity.
The most useful sets of orthogonal functions for our purposes
are the wave functions belonging to a given wave equation. In
preceding chapters we have shown that the solutions of certain
wave equations form sets of normalized orthogonal functions,
such as for example the Hermite orthogonal functions which
154 PERTURBATION THEORY [VI22
are the solutions of the harmonic oscillator problem (Sec. 11).
In Appendix III it is shown that the solutions of any wave
equation form such a set of orthogonal functions.
In making expansions in terms of orthogonal functions, it is
necessary to be sure that the set of junctions is cam,pLeLe. Thus in
the example of Equations 224 and 225, if we had used the set
cos x, cos 2x, • • • , without the sine terms, the series obtained
would have converged, but not to the function <p(x), because the
set of functions cos x, cos 2x, • • • is not complete. This
requirement of completeness necessitates that all the solutions
of the wave equation be included when using these solutions for
an expansion of an arbitrary function. Since many wave equa
tions lead to a continuous spectrum of energy levels as well as a
discrete spectrum, it is necessa ry to incl ude the wave f unction^
belonging 4o the continuous levels when making an expansion.
The quantum numbers for the continuous spectra do not have
discrete values but may vary continuously, so that the part
of the expansion involving these wave functions becomes an
integral instead of a sum as in Equation 221.
However, in many special cases it is easy to see that certain
of the coefficients a* will be zero so that in those cases an expan
sion is possible in a set of functions which is not complete. Thus
if the function <p(x) which we are attempting to represent is an
even function 1 of x y and if the orthogonal set we are using for the
expansion contains both even and odd functions, the coefficients
of all the odd functions fk{x) will vanish, as may be seen from the
consideration of Equation 222.
All the ideas which have been discussed in this section can be
generalized without difficulty to systems of several variables.
Normalized orthogonal functions in several variables Xi, yi,
• • • , z N satisfy the condition *
/ " • " !fi(*u Vu ' " ' f **)/m(xi, 2/i, • • • , z N )dr
Oifn*m,i (22 _ 6)
= 1 if n = m, )
in which the integration is carried out over the whole of the
configuration space for the system, and dr is the volume element
1 The function f(x) is called an even function of x if f(—z) is equal to
f(x) for all values of x, and an odd function of x if /( — x) is equal to ~/(a?)
for all values of x.
VI22] EXPANSIONS IN SERIES OF ORTHOGONAL FUNCTIONS 155
for the particular coordinate system in which the integral is
expressed. Orthogonal functions in several variables usually
are distinguished by several indices, which may however be
symbolized by a single letter. An example of a threedimensional
set of normalized orthogonal functions is the set of solutions of
the wave equation for the hydrogen atom. We have obtained in
Chapter V the solutions belonging to the discrete levels; the
quantum numbers nlm provide the indices for these functions.
The solutions for the continuous spectrum of the atom, i.e., the
system resulting when the electron has been completely removed
from the nucleus, must be included if a complete set is desired. 1
The coefficients in the expansion of an arbitrary function of
several variables are obtained from an equation entirely analogous
to Equation 222,
a* = / • • • JVtei, Vu * * ' , z*)/*(zi, Vu ' ' ' , Zn)<It } (227)
in which the limits of integration and the meaning of dr are tne
same as in Equation 226.
A function <p which is expressed in terms of the normalized
functions of a complete orthogonal set is itself normalized if the
coefficients in the "expansion satisfy the relation 2ja*a n = 1.
n
It may be mentioned that in some cases it is convenient to
make use of complete sets of functions which are not mutually
orthogonal. An arbitrary function can be expanded in terms
of the functions of such a set; the determination of the values
of the coefficients is, however, not so simple as for orthogonal
functions. An example of an expansion of this type occurs in
Section 24.
In certain applications of expansions in terms of orthogonal
functions, we shall obtain expressions of the form
Xa n fn(x) = 0.
n
By multiplying by /*(x) and integrating, we see that the coeffi
cient of each term must be zero; i.e., a n = for all values of n.
1 For a discussion of the wave functions for the continuous spectrum of
hydrogen, see Sommerfeld, "Wave Mechanics," p. 290.
156 PERTURBATION THEORY [VI23
Problem 221. Obtain the first four coefficients in the expansion <p(x) =
(2T + *)H = ^ ak yJ^^P k (x) f where P k (x) is the fcth Legendre poly
nomial given in Section 19. This expansion is valid only for \x\ $ 1. Plot
ip(x) and the approximations to it given by including the first, second, third,
and fourth terms of the expansion. If possible, obtain a general expression
for an, using the generating function for Pk(x).
^3. FIRSTORDER PERTURBATION THEORY FOR A
NONDEGENERATE LEVEL
In discussing many problems which cannot be directly solved, a
solution can be obtained of a wave equation which differs from
the true one only in the omission of certain terms whose effect
on the system is small. Perturbation theory provides a method
of treating such problems, whereby the ap proximate eq uation
js first solved and th gnjji e sm a^additional terms are introduced
as correcti ons^
Let us write the true wave equation in the form
m  W+ = 0, (231)
in which H represents the operator
H = 8.
?2i* + F  (23  2)
We assume that it is possible to expand H in terms of some
parameter X, yielding the expression
H = H° + \H' + X 2 #" + • • ' 1 (233)
in which X has been chosen in such a way that the equation to
which 231 reduces when X — > 0,
#<y,o _ W o r = 0> (23_4)
can be directly solved. This equation is said to be the wave
equation for the unperturbed system, while the terms
X#' + X 2 #" + • • •
are called the perturbation. As an illustration, we might men
tion the problem of the Stark effect in atomic hydrogen, in
which an electric field is applied to the atom. In this problem
the field strength E provides a convenient parameter in terms
VI23] FIRSTORDER PERTURBATION THEORY 157
of which the Hamiltonian may be expanded. When E is zero,
the problem reduces to that of the ordinary hydrogen atom,
which we have already solved.
The unperturbed equation 234 has solutions
+1, K K • • • , *2, • • • ,
called the unperturbed wave functions, and corresponding energy
values
wi m w% • • • , wi,  • • .
The functions $! form a complete orthogonal set as discussed in
Section 22, and, if we assume that they have also been normalized,
they satisfy the equation (Appendix III)
WW
T = o if i * j\
= i if i = j.f
(235)
Now let us consider the effect of the perturbation. By hypoth
esis it will be small, and from the continuity properties of wave
functions 1 we know that the energy values and wave functions
for the perturbed system will lie near those for the unperturbed
system. In other words, the application of a small perturbation
is not going to cause large changes. With these facts in mind
we can expand the energy W and the wave function \p for the
perturbed problem in terms of X and have reasonable assurance
that the expansions will converge, writing
fa = V k + WL + XVr + • • • (236)
and
fa = Wl+ ^l + VW k ' + ' * ' . (237)
If the perturbation is really a small one, the terms of these series
will become rapidly smaller as we consider the coefficients of
larger powers of X; i.e., the series will converge.
We now substitute these expansions for H> fa, and W k into
the wave equation 231, obtaining the result, after collecting
coefficients of like powers of X,
(H«V k  Wl+l) + {H°+' k + H'+l  WM  WM)\
+ (H%' + H'i' k + H"W  WW  WM  W'M)\*
+ • • • =0. (238)
1 Discussed, for example, in Courant and Hilbert, " Methoden der mathe
matischen Physik."
158 PERTURBATION THEORY [VI23
If this series is properly convergent, we know that in order for it to
equal zero for all values of X the coefficients of the powers of X
must vanish separately. 1 fyhe coefficient of X° when equated to
zero gives Equation 234, ip that we were justified in beginning
the expansions 236 and 23^7 with the terms \p° and W°. The
coefficient of X gives the equation
H°+' k  Wfr' k = (W' k  ff')«. (239)
To solve this we make use of the expansion theorem discussed
in the last section. We consider that the unknown functions
\f/ f k can be expanded in terms of the known functions ^?, since the
latter form a normalized orthogonal set, and write
ti = X a ^°' (2310)
i
(The coefficients a t might be written as an, but we shall assume
throughout that we are interested only in the state k and there
fore shall omit the second subscript.) Using this, we obtain the
result
i i
since
tf<ty? = WW.
Equation 239 therefore assumes the form
XatiW*,  WWf = (W' k  H'M. (2312)
If we multiply by ypl * and integrate over configuration space, we
observe that the expression on the left vanishes :
J*8*2ai(W7  WMdr *> Joi(lF? WDM **?&■ = 0,
i I
since JV£ *tfdr vanishes except for I = k t and for this value
1 Thus, if
^a n X» ss *>(x) s 0,
n
then, assuming that the series is properly convergent, we can write
M^)  o.
n!\d\ tt /x~o
VI23] FIRSTORDER PERTURBATION THEORY 159
of I the quantity W? — W k vanishes; and hence we obtain the
equation
ir(ir;ff# = o. (2313)
This solves the problem of the determination of W k , the first
order correction to the energy. Since W k is a constant in
Equation 2313, the integration of the term containing it can be
carried out at once, giving the result, when multiplied by X,
W k =\M*HW k dr. (2314)
Since the correction to the energy is \W f k , it is convenient to
include the parameter X in the symbols for the firstorder pertur
bation and the firstorder energy correction, so that to the
first order it is usual to write the relations
H = H° + H' y )
W k = W° k + W'J
in which
W' k = SH *H'+ldr. (2316)
This expression for the perturbation energy can be very simply
described : The firstorder perturbation energy for a nondegenerate
state of a system is just the perturbation function averaged over the
corresponding unperturbed state of the system.
We can also evaluate the correction \f/' k for the wave function.
Multiplying each side of Equation 2312 by tf *, we obtain, after
integration^
a,(W?  Wl) = /*? *#Wr, j ^ k, (2317)
where we have utilized the orthogonality and normalization
properties of the ^ 0, s. The coefficients a ; in the expansion 2310
of yp' in terms of the set ip? are thus given by the relation
■VW^W' i* (2318,
The value of a k is not given by this process; it is to be chosen
so as to normalize the resultant \p, and, if only firstorder terms
are considered (terms in X 2 neglected), it is equal to zero. It is
convenient to introduce the symbol
H' ik = M*H'fflT, (2319)
160 PERTURBATION THEORY [VI23a
so that the expression for the firstorder wave function of the
system, on introducing the above values of the coefficients a,,
becomes
00
*» = *! x 2 V?Vg *°' (23 ~ 20)
yo
in which the prime on the summation indicates the omission of
the term with j = k.
As mentioned before, it is customary to include X in the defini
tion of H' as indicated in Equation 2315, so that we get finally
for the firstorder energy and the firstorder wave function the
expressions
W k = Wl + H' kk (2321)
and
oo
+» = +1  S' w7V ' 9  (23_22)
23a. A Simple Example : The Perturbed Harmonic Oscillator.
As a simple illustration of firstorder perturbation theory we shall
obtain the approximate energy levels of the system whose wave
equation is
dx*
+ ^™(w  \kx*  ax*  bx*\ = 0. (2323)
We recognize that if a and b were zero this would be the wave
equation for the harmonic oscillator, whose solutions we already
know (Sec. 11). If a and b are small, therefore, we may treat
these terms as perturbations, writing
H' = ax* + bx\ (2324)
We need then to evaluate the integrals
H '*n = a f^y°n * X W dx + 6 /.V ^ **WdX. (23"25)
Since x 8 is an odd function and \pl *\pl an even function, the first
of these integrals is zero, so that the firstorder perturbation due
to ay? is zero. To calculate the second integral we refer back to
V23a] FIRSTORDER PERTURBATION THEORY 161
Section lie for the functions ^° and their properties. Substi
tuting for xpl from Equation 1120 we obtain the integral
1 = j_\+»* x w dx = SX/" i2 ^ (t) ^^ (23_26)
From Equation 1115 we see that
fff.({) = ^H» +1 (f) + tt#ni(a (2327)
so that, after applying Equation 2327 to %H n +x and f// n i and
collecting terms, we obtain the equation
eHn(Z) = KHn+tiZ) + {n + K)ffn(«) + n( W  l)ffn 2 (f).
(2328)
By this application of the recursion formula for H n (£) we have
expressed £ 2 H n (£) in terms of Hermite polynomials with constant
coefficients. By squaring this we obtain an expression for
£ A Hl(£), which enables us to express the integral in Equation
2326 as a sum of integrals of the form
f~j* t H n (t)H m (Z)dl; = if m * nl (23 _ 2 9)
= 2 n n\yfic if m = n,)
evaluated in Section lie. Thus we find for I the expression
1 = ^^{iV" +2(n + 2)! + ( n + ii 2nni +
w 2 (n  l) 2 2" 2 (n  2)\>
when the value of N n given in Equation 1121 is introduced.
The firstorder perturbation energy for this system is therefore
W = H' M = ^ 2 (2n> + 2n + 1),
so that the total energy becomes (to the first order)
W = W° + W'=(n + ^jhp + ~(2n + 2n + 1)^
(2330)
162 PERTURBATION THEORY [VT23b
In order to calculate the firstorder wave function it would
be necessary to evaluate all the quantities H ' nk . The x z term as
well as the x 4 term will contribute to these integrals. The
number of nonzero integrals is not, however, infinite in this
case but quite small, only the terms with k = n, n ± 1, n ± 2,
n ± 3, and n ± 4 being different from zero.
23b. An Example: The Normal Helium Atom. — As another
example of the application of firstorder perturbation theory let
us discuss the normal state of the helium atom. Since the term
which we shall use as the perturbation is not particularly small,
we must not expect an answer of very great accuracy. The
potential energy for a system of two electrons and a nucleus of
charge +Ze is
Ze 2 Ze 2 p 2
F= — t±+±, (2331)
ri r 2 r i2
in which r x and r 2 are the distances of electrons 1 and 2, respec
tively, from the nucleus, and r J2 is the separation of the two
electrons. If we make the approximation of considering the
nucleus at rest, which introduces no appreciable error, the wave
equation (see Equation 128) for the two electrons becomes
H + = 
&r 2 m
/ ay ,^,^,^,!?V. <W\
\dx\ + dy\ + dz\ + dx\ + dy\ + dz\)
( Ze 2 Ze 2 e 2 \
+ ( T~ t+^)+ = W+. (2332)
This equation applies to He, Li+, Be" 1 " 1 ", etc., with Z = 2, 3, 4, etc.,
respectively. The variables x u 2/1, Z\ are Cartesian coordinates
of one electron, and x 2 , t/ 2 , z 2 those of the other; ra is the mass
of the electron.
Since if the term 6 2 /r i2 is omitted the wave equation which
is obtained can be exactly solved, we choose this term as the
perturbation function,
p 2
H' =— 
The wave equation which remains, the unperturbed equation,
can then be separated into two equations by the substitutions
^°(zi, 2/1, *i, s 2 , 2/2, z 2 ) = u\(x h y h Zi)ul(x* } 2/ 2 , z 2 )
VI23b] FIRSTORDER PERTURBATION THEORY 163
and
w° = w\ + W° 2J
the equation 1 for u\ being
d 2 u\ dV x dhjj 8whn 0i
dx\ + dy\ + dz\ + A* ' '
/^o + ^£_ 2 j w o = (23 _3 3)
The equation for w§ is identical except for the changed subscripts.
Equation 2333 is just the hydrogenlike wave equation discussed
in the preceding chapter, with solutions ^ n im{ri 9 #i> <Pi) and
energy values — Z 2 W H /n 2 } in which
27r 2 m 6 4 1Q KQ
?/? = — p — = 13.53 v.e.
The unperturbed wave function for the lowest level of the two
electron atom is therefore
^100,100 = ^ioo(ri, #i, ^i)^ioo(r 2 , #2, <Pi) =
u u (r h #1, <pi)u u (r 2 , #2, <p 2 ), (2334)
in which r\ } #1, <pi and r 2 , #2, ^2 are polar coordinates of the two
electrons relative to axes with the nucleus at the origin. The
corresponding energy value is
TFSoo.100 = Wl + W\ = 2Z 2 W H . (2335)
The firstorder perturbation energy W is the average value of
the perturbation function H f = e 2 /r n over the unperturbed
state of the system, with the value
W = Jr*HWdr = j£*ioo.ioodr. (23
36)
From Table 214 of Chapter V we obtain for u u the expression
tii. = *ioo = xffie"*, (2337)
\7ra
in which p = 2Zr/a and <z = h 2 /±T 2 m e 2 . Using this in Equa
tion 2334, we find for ^100,100 the expression
^100,100 = — z e 2 e 2 .
TTUq
1 The symbol u will be used for the wave function for a single electron in a
manyelectron atom, with subscripts Is, 2s, 2p, etc.
164 PERTURBATION THEORY [VI23b
The volume element is
dr = r\dri sin &xd&id(pi • rfdr 2 sin # 2 d*M<p2,
so that the integral for W becomes
Ze 2
W =
9b »  — — Pi^Pi sin 0,
* 7ra Jo Jo Jo Jo Jo Jo P12
d$id<pipldp 2 sin # 2 d# 2 (i<p 2 , (2338)
in which p i2 = 2Zri 2 /a .
The value of this integral is easily obtained, inasmuch as it
corresponds to the electrostatic interaction energy of two spheri
cally symmetrical distributions of electricity, with density
functions e~ pi and e~ P2 , respectively. In Appendix V it is shown 1
that
W' = */iZW H . (2339)
This treatment thus gives for the total energy the value
W = (2Z 2  %Z)W«. (2340)
This may be compared with the experimental values of the total
energy, which are obtained by adding the first and the second
ionization energies. Table 231 contains, for He, Li + , Be ++ ,
B 3 +, and C 4_l_ , the experimental energy W exp ., the unperturbed
energy W°, the total energy calculated by firstorder perturbation
theory W° + W, the difference A = W e x V .  W°, the difference
A' = W^p.  W°  W, and finally the ratio A'/A .
It is seen that the error A' remains roughly constant in absolute
value as the nuclear charge increases, which means that the
percentage error decreases, since the total energy is larger for
larger Z. This result is to be expected, inasmuch as for large
nuclear charge the contribution of the attraction of the nucleus
is relatively more important than that of the repulsion of the two
electrons. It is pleasing that even in this problem, in which the
perturbation function e 2 /r u is not small, the simple firstorder
perturbation treatment leads to a value of the total energy of
the atom which is in error by only a small amount, varying from
5 per cent for He to 0.4 per cent for C 4+ .
1 This problem was first treated by A. Unsold, Ann. d. Phys. 82, 355
(1927).
VI24]
FIRSTORDER PERTURBATION THEORY
165
Table 231. — Calculated and Observed Values of the Energy of
Heliumlike Atoms and Ions
W txp .,v.e.
JP°, v.e.
W°W',
v.e.
A , v.e.
A', v.e.
A'/A°
He
78.62
108.24
74.42
29.62
4.20
0.142
Li+
197.14
243.54
192.80
46.40
4.34
.094
Be + +
369.96
432.96
365.31
63.00
4.65
.074
B 3+
596.4
676.50
591.94
80.1
4.5
.056
C« +
876.2
974.16
872.69
98.0
3.5
.036
Problem 231. Calculate the firstorder energy correction for a one
dimensional harmonic oscillator upon which the perturbation H'{x) acts,
where H'(x) is zero unless x < e and H'(x) = b for \x\ < e, with e a quantity
which is allowed to approach zero at the same time that b approaches infinity,
in such a way that the product 2eb = c. Compare the effect on the odd and
even levels of the oscillator. What would be the effect of a perturbation
which had a very large value at some point outside the classically allowed
range of the oscillator and a zero value elsewhere?
Problem 232. The wave functions and energy levels of a particle in a
onedimensional box are given in Equations 146 and 14r7. Calculate the
firstorder perturbation energy for such a system with a perturbation H'
such that H' = b for (a/A;)  t ^ x ^ {a/k) f e and H' = elsewhere, with
€ — » as b — ► oo in such a way that 2eb = c, k being a given integer. With
k = 5, determine which energy levels are the most and which are the least
perturbed and explain. With k = 2, give the expression for the perturbed
wave function, to the first order.
Problem 233. Let H' be a perturbation, such that H'(x) = —b for
^ x ^ a/2 and H'{x) ■= +b for a/2 ^ x ^ a, which is applied to a
particle in a onedimensional box (Eqs. 146 and 147). Obtain the first
order wave function. Show qualitatively that this function is such that the
probability of finding the particle in the righthand half of the box has been
increased and explain in terms of classical theory. (Hint: Use the symmetry
about the point x = a/2.)
24. FIRSTORDER PERTURBATION THEORY FOR A DEGENERATE
LEVEL
The methods which we have used in Section 23 to obtain the
firstorder perturbation energy are not applicable wjien the energy
level of the unperturbed system is degenerate/for the reason
that in carrying out the treatment we assumed that the perturbed
wave function differs only slightly from one function ^ which
is the solution of the unperturbed wave equation for a given
energy value whereas now there are several such functions, all
166 PERTURBATION THEORY [VI24
belonging to the same energy level, and we do not know which
one (if any) approximates closely to the solution of the perturbed
wave equation.
An energy level Wk is called afold degenerate (see Sec. 14)
when for W = Wk there exist a linearly independent wave func
tions \pk\f 4 / k2 f ^*3, • • • , ^ka satisfying the wave equation. 1
Each of these is necessarily orthogonal to all wave functions for
the system corresponding to other values of the energy (see
Appendix III) but is not necessarily orthogonal to the other
functions corresponding to the same value of the energy. Any
a
linear combination ]$►* Kj^/ k j of the wave functions of a degenerate set
yi
such as ypki y ^*2, • • • , ^k<x is itself a solution of the wave equa
tion and is a satisfactory wave function corresponding to the
energy IF*. We might therefore construct a such combinations
Xk% by choosing sets of values for k, such that the different com
bin ations thus formed are linearly independent. "The^setrtjf
functions so obtained,
a
Xki = 2) *f**/, t = 1, 2, 3, ■ ■ • , a, (241)
is entirely equivalent to the original set \p k i, ^*2, • • • , ^*«.
This indicates that there is nothing unique about any particular
set of solutions for a degenerate level, since we can always con
struct an infinite number of other sets, such as x*i> • • • > Xka,
which are equally good wave functions. The transformation
expressed by Equation 241 is called a linear transformation
with constant coefficients.
It is usually convenient to deal with wave functions which
are normalized to unity and which are mutually orthogonal
Since the coefficients *»/ can always be chosen in such a way as
to make the set x*» possess these properties, we shall ultimately
assume that this has been done.
Using these ideas, we can now investigate the application of
perturbation theory to degenerate levels. We write the wave
equation in the form
1 The functions \pki, \frkt, • • • , tha are said to be linearly independent if
there exists no relation of the form a^ki + aii>k% + • • • + Ocrtka = (in
which a\, a t , • • • , a a are constant coefficients) which is satisfied for all
rallies of the independent variables.
VI24] FIRSTORDER PERTURBATION THEORY 167
Hi  W* = (242)
with
H = H« + \H' + X 2 #" + • • •
as before. The wave equation for the unperturbed system is
#y>  W°V = 0, (243)
the solutions of which are
*Si, * o2, • • • ; *Si, *Si, •••;•••; *fc, *&, • • • ,
«; • • • ;
corresponding to the energy levels
Wl W\; ■ ■ ■ ; Wli • • • •
Now let us consider a particular wave function for the per
turbed equation 242. It is known, in consequence of the proper
ties of continuity of characteristicvalue differential equations,
that as the perturbation function \H' + • • • becomes smaller
and smaller the energy value W of Equation 242 will approach
an energy level of the unperturbed equation 243, Wl, say.
The wave function under consideration will also approach more
and more closely a wave function satisfying Equation 243.
However, this limiting wave function need not be any one of
the functions ^lu • • * > ^L> ft ma y be (and generally is) some
linear combination of them. The first problem which must be
solved in the treatment of a degenerate system is the determina
tion of the set of unperturbed wave functions to which the
perturbed functions reduce when the perturbation vanishes;
that is, the evaluation of the coefficients in the linear transforma
tion converting the initially chosen wave functions into the
correct zerothorder wave functions. These correct combinations,
given by
xli = X *»'*&'' *  1, 2, • • • , a, (241)
J'«i
provide the first term of the expansion of \f/ki in powers of X, since
by definition they are the functions to which the \puis reduce when
X > 0. Therefore
*« = xli + Wki + XVii + • • ' (245)
and
Wki = Wt + \W' kl + W& + • • • , (24r^)
168 PERTURBATION THEORY [VI24
where l(— 1, 2, • • • , a) designates the particular one of the a
degenerate wave functions in question, are the equations which
are analogous to Equations 236 and 237. (As in Equation
2310 we sometimes omit the subscript k; e.g., we write w
for Kkiv) it must be borne in mind that throughout we are con
sidering the fcth degenerate level.)
Substituting the expansions for \{/, W, and H into the wave
equation 242, we obtain an equation entirely analogous to
Equation 238 of the nondegenerate treatment,
{H\l t  TF2x2i) + (HVi, + ff'xJi  ww kl  WM)X +
• • • = 0, (247)
from which, on equating the coefficient of X to zero as before,
there results the equation (cf. Eq. 239)
Wh  W° k r H = WUxlt  H'xl (248)
So far our treatment differs from the previous discussion of
nondegenerate levels only in the use of xu instead of \p ki ) i.e.,
in the introduction of a general expression for unperturbed
functions instead of the arbitrary set \p ki . In the next step we
likewise follow the previous treatment, in which the quantities
\l/' k and H°\p k were expanded in terms of the complete set of
orthogonal functions \f/ k . Here, however, we must in addition
express x& in terms of the set \fr kl >, by means of Equation 244,
in which the coefficients k iV are so far arbitrary. Therefore we
introduce the expansions
tii = XawrMv (240)
kT
and
flV« = XawrHWr = %a kl k>i>WW k , v (2410)
h'V k'V
into Equation 248 together with the expression for xli given by
Equation 244. The result is
a
k'v ri
in which the righthand side involves only functions \// kV belonging
to the degenerate level Wl while the expansion on the left includes
VI24] FIRSTORDER PERTURBATION THEORY 169
all the $j*>i>'8. If we now multiply both sides of this equation by
4*1* and integrate over configuration space, we obtain the result
a
MtH'Mvdr). (2412)
The left side of this equation is zero because \f/ ki and $j r are
orthogonal if k ^ fc' and TTg.  F£ is zero if fc = fc'. If we
introduce the symbols
E' iV = MfH'^dr (2413)
and
**v = R%°A, (2414)
we may express Equation 2412 in the form
2V(#Jr ~ A ? rF^) =0, j = 1, 2, 3,  • • , a. (2415)
ri
This is a system of a homogeneous linear simultaneous equations
in the a unknown quantities kh, ki 2 , • • • , k&*. Written out in
full, these equations are
{H' u  &nW' kl )Kn + (H[ 2  A lt WU)*it + ' ' ' +
(H' la  A la TTi,)ic te = > ]
(ff 21  A 2 i^)k Z i + .(# 22  A 22 J^)k I2 + • • • +
(# 2a  AtoWMicta = 0,> (2416)
(H' al  AaiW' kl )Kn + (£Ti s  A a ,W' kl )K l2 + * * * +
Such a set of equations can be solved only for the ratios of the
k's; i.e., any one k may be chosen and all of the others expressed
in terms of it. For an arbitrary value of W kU however, the set
of equations may have no solution except the trivial one kw — 0.
It is only for certain values of W' kl that the set of equations has
nontrivial solutions; the condition that must be satisfied if
such a set of homogeneous linear equations is to have nonzero
solutions is that the determinant of the coefficients of the
unknown quantities vanish ; that is, that
170
PERTURBATION THEORY
[VI24
H' n  AulPS, H' u  A u W' kl
H'nAuWU H' n A 2i W' kl
H' la  A la W' kl
H' ta  AtoW'u
H' al A al W' kl HiaA^W'u
"act &aaW k i
= 0.
(2417)
This determinantal equation can be expanded into an algebraic
equation in W' hl which can then be solved for W ki . For the types
of perturbation functions which arise in most physical and chemi
cal problems the determinant is either symmetrical about the
principal diagonal, if the elements are real, or else has the property
that corresponding elements on opposite sides of the principal
diagonal are the complex conjugates of each other; that is,
H' {i = H'f. In consequence of this property it can be shown
that the determinant possesses a real roots, W' kl , W k2 , • • • , W' ka 
These are the values of the firstorder perturbation energy for the
a wave functions which correspond to the afold degenerate
unperturbed energy level W k . It may happen, however, that
not all of the roots W kl , etc., are distinct, in which case the
perturbation has not completely removed the degeneracy.
The coefficients kh> which determine the correct zerothorder
wave function xli corresponding to any perturbed level W kt
may be determined by substituting the value found for W kl
into the set of simultaneous equations 2416 and solving for the
other coefficients in terms of some one of them. This remaining
arbitrary coefficient may be adjusted so as to normalize xli
This process does not give uriique results if two or more roots W k i
coincide, corresponding to the fact that since there still remains
a certain amount of degeneracy the wave functions for the
degenerate level are not uniquely determined but are to a
certain degree arbitrary.
If the original wave functions fin ' ' ' t W« were normalized
and mutually orthogonal (which we have not hitherto needed
to assume), the function Ayr is unity for j = V and zero other
wise, so that the determinantal equation 2417 assumes the
form
#n — W k 'i H'n H[ z
H' 21 H' 2% — W' M #«
H ai
H**
Hal
H ia
H 2 a
HL  W' kl
= 0.
(2418)
VI24] FIRSTORDER PERTURBATION THEORY 171
An equation such as 2417 or 2418 is often called a secular
equation, and a perturbation of the type requiring the solution
of such an equation a secular perturbation. 1
It is interesting to note that in case the secular equation has
the form
H' n  W' kl
H 22 W kl
H'aaWL
0,
(2419)
then the initially assumed functions yp kx , \l/ k2 , • • • > Vka are the
correct zerothorder functions for the perturbation H', as is seen
on evaluation of the coefficients k of Equations 2416. A secular
equation in which all the elements are zero except along the
principal diagonal is said to be in diagonal form. The roots
W' kl are of course immediately obtainable from an equation in
this form, since the algebraic equation equivalent to it is
(ffii " W' kl )(H' 22  W' kl )   • (HL  W' kl ) = 0, (2420)
with the roots W' kl = H' u , H 22y • • • , H' aa .
1 In this sense secular means "accomplished in a long period of time"
(Latin saecidum = generation, age). The term secular perturbation was
introduced in classical mechanics to describe a perturbation which produces
a slow, cumulative effect on the orbit. If a system of sun and planet, for
which the unperturbed orbits are ellipses of fixed size, shape, and orientation,
were perturbed in such a way as to change the law of force slightly from the
inverse square, as is done, for example, by the relativistic change of mass
with change of speed, the position of the major axis in space would change
by a small amount with each revolution of the planet, and the orbit would
carry out a slow precession in its own plane, with a period which would be
very long if the magnitude of the perturbation were small. Such a perturba
tion of the orbit is called a secular perturbation.
On the other hand we might have a system composed of a wheel in a
gravitational field rotating about a horizontal frictionless axle passing
through its center of mass and perturbed by the addition of a small weight
at some point on its periphery in such a way as to accelerate the motion
as the weight moves down and to decelerate it as the weight moves up.
Such a perturbation, which produces a small effect on the motion with the
high frequency characteristic of the original unperturbed motion of the
system, is not a secular perturbation.
The significance of the use of the word secular in quantum mechanics
will be seen after the study of the perturbation theory involving the time
(given in Chap. XI).
172
PERTURBATION THEORY
[VT24a
This equation 2419 illustrates, in addition, that the integrals
H' mn depend on the set of zerothorder functions ^li which is
used to define them. Very often it is possible to guess in advance
which set of degenerate ^ 0, s to use for a given perturbation in
order to obtain the simplest secular equation. In particular,
in case that the perturbation is a function of one variable (x, say)
alone, and each function of the initial set of unperturbed wave
functions can be expressed as the product of a function of x and a
function of the other variables, the individual functions being
mutually orthogonal, then these product functions are correct
zerothorder wave functions for this perturbation. This situa
tion arises whenever the unperturbed wave equation can be
separated in a set of variables in which x is included.
It may be pointed out that Equation 2418 may also be written
in the form
ffn  W H l2
#21 #22 ~ W
Hex l
H a
H\a
H 2 a
H aa  W
0,
in which Ha = H% + H^ and W = W° k + W' kl , inasmuch as
//§ is equal to Wl for i = j and to zero for i ^ j. This form is
used in Section 30c.
24a. An Example : Application of a Perturbation to a Hydrogen
Atom. — As an illustration of the application of perturbation
theory to degenerate systems, let us consider a hydrogen atom to
which a perturbation which is a function of x only has been
applied. Since the lowest state of the hydrogen atom is non
degenerate, the treatment of Section 23 applies to it and we have
the result that
W = Moof(x)dr
with W = f(x). For the second energy state, however, we need
to use the treatment for degenerate systems, since for W\ = — %
Rhc there are four wave functions,
^2. = ^20
fa* = ^211
\32raJ \a J
cost?,
VI24a] FIRSTORDER PERTURBATION THEORY 173
**  *  aE 6 "" © • ^ e_i * sin *'
** = **»  >&'^) • ^ e+ * sin *»
as given in Chapter V. In order to set up the secular equation
for this system we need the integrals
Even without specifying the form of the function /(x) further,
we can say certain things about these integrals. Since the com
plex conjugate of e~~ lV is e+*> and e~ t >e +t "*' = 1, we see that
" = "2lT,2lI = "211,211
regardless of the nature of H', so long as it is real. By expressing
x in polar coordinates through the equation
x = r sin # cos <p,
we see that f(x) is the same function of <p' = 2tt — <p as it is of <p,
since cos (2tt — p) = cos <p. If we make this substitution in
an integral over <p we get the result
f 2 "g(<p)d<p = £j(2* ~ <p'W = J^VZar  ^W =
ffifrr  v)d<p, (2421)
since it is immaterial what symbol we use for the variable of
integration in a definite integral. This substitution also changes
e~ i<p into e** 2 *^ or e + * v ', so that by its use we can prove the
identity
D = "200.2U == "200,211
f(x) is also unchanged in form by the substitution & = tt •— #',
since sin (x — #') = sin #'. Also, we have the relation
fVW sin &d$ = fW  #') sin #'<M' =
r fif(x  t>) sin tfcW, (2422)
X'
/o
in which the factor sin # is introduced because it occurs in the
volume element dr of polar coordinates. The substitution
# = 7T — $' does not leave cos & unchanged, however, since
cos (it — #') = — cos #'. By employing this substitution we
can show that
174
PERTURBATION THEORY
[VI24a
"210,200 — """210,200
or
ffsi
= 0,
'210,200
since the integrand is unaltered by the substitution except for
the cosine factor in ^§ 10 which changes sign. Similarly we
find
"210,211 "~
and Ho
210,211
= 0.
Finally we have the general rule that
TJ f — TT f *
n 2lm,2Vm' ~ " 2l'm',2ltn'
We are now in a position to write down the secular equation for
this perturbation, using the relations we have obtained among
the elements H' tlmtiVm >. It is (using the order 200, 211, 211, 210
for the rows and columns)
AW D D
D B W E
D E B  W
CW
0. (2423)
The symbols A, B, etc., have the meanings: A = # 20
B — /i2ii,211> C — H 2
10,210)
D =H'
and E = HL
We may obtain one root of this equation at once. Since the
other elements of the row and the column which contains C — W
are all zero, C — W is a factor of the determinant and may be
equated to zero to obtain the root W = C. The other three
roots may be obtained by solving the cubic equation which
remains, but inspection of the secular equation suggests a simpler
method. Determinants have the property of being unchanged
in value when the members of any row are added to or sub
tracted from the corresponding members of any other row. The
same is true of the columns. We therefore have
A 
W
t
D
D
D
B  W
E
D
E
B W
1
AW
2D
"2
D B
D B
W'+E B W  E
W' + E E  B + W
1
A
—
W 2D
1
~4
2D
2(5 + E 
W)
= C
2(5  1
E  W)
(2424)
VI24a] FIRSTORDER PERTURBATION THEORY 175
in which we have first added the last column to the second
column to form a new second column and subtracted the last
column irom the second column to form a new third column, and
then repeated this process on the rows instead of the columns.
The result shows that we have factored out another root,
W' = B — E, leaving now the quadratic equation
(A  W'KB + E  W)  2Z> 2 =
which determines the remaining two roots.
The process by which we have factored the secular equation
into two linear factors and a quadratic corresponds to using
the real functions ^ 2 „ ^2 Px , fap,, and \f/ 2 p M for the ^'s instead
of the set ^ 2 „ ^ 2P i, ^2 Pl , and ^ 2po (see Sec. 186). In terms of
the real set the secular equation has the form
A  W V2D
V2D B + E  W
B  E  W
C  W
= 0,
(2425)
which, aside from the last row and column, differs from the last
determinant of Equation 2424 only by a constant factor. The
proper zerothorder wave functions for this perturbation are
therefore ^ 2p „ ^ 2po and two linear combinations a^ 2 , + W 2p ,
and $yp2* — «^ 2px , in which the constants a and ft are determined
by solving the quadratic factor of the secular equation, sub
stituting the roots into the equations for the coefficients of the
linear combinations, and solving for the ratio a/fi. The
normalization condition yields the necessary additional equation.
It is to be noted that in place of ^ 2p „ and ^ 2p , any linear com
binations of these might have been used in setting up the secular
equation 2425, without changing the factoring of that equation,
so that these linear combinations would also be satisfactory
zerothorder wave functions for this perturbation.
Problem 241. Prove the statement of the last paragraph.
Problem 242. Discuss the effect of a perturbation f(y) [in place of /(x)]
on the system of Section 24a.
176 PERTURBATION THEORY [VI25
25. SECONDORDER PERTURBATION THEORY
In the discussion of Section 23 we obtained expressions for
W and V in the series
W = W° + \W + XW + • • • (251)
and
+ = xfr° + W + XV" + (252)
In most problems it is either unnecessary or impracticable to
carry the approximation further, but in some cases the second
order calculation can be carried out and is large enough to be
important. This is especially true in cases in which the first
order energy W is zero, as it is for the Stark effect for a free
rotator, a problem which is important in the theory of the meas
urement of dipole moments (Sec. 49/).
The expressions for W" and ^" are obtained from the equation
which results when the coefficient of X 2 in Equation 238 is put
equal to zero and a solution obtained in a manner similar to
that of the firstorder treatment. We shall not give the details
of the derivation but only state the results, which are, for the
energy correction,
in which
and
B' u = SWH'Mdr (254)
HU = WH'Wdr (255)
and the prime on 2 means that the term I = k is omitted. All
other values of I must be included in the sum, however, including
those corresponding to the continuous spectrum, if there is one.
If the state W% is degenerate and the firstorder perturbation
has removed the degeneracy, then the functions to be used in
calculating H' kl , etc., are the corrects zerothorder functions found
by solving the secular equation.
If the energy level for the unperturbed problem is degenerate
and the firstorder perturbation does not remove the degeneracy >
the application of the secondorder correction will also not remove
the degeneracy unless the term X 2 jff" is different from* zero, in
VI25a] SECONDORDER PERTURBATION THEORY 177
which case the degeneracy may or may not be removed. The
treatment in this case is closely similar to that of Section 24.
25a. An Example : The Stark Effect of the Plane Rotator.— A
rigid body with a moment of inertia I and electric moment 1 n, con
strained to rotate in a plane about an axis passing through its
center of mass and under the influence of a uniform electric field
E, is characterized by a wave equation of the form 2
0+^OF + Kfl COS *)*=<>,
in which <p is the angle of rotation. If we call — y.E cos <p the
perturbation term, with E taking the place of the parameter X,
then the unperturbed equation which remains when E = has
the normalized solutions
r m = 7^ eim *> ^ = 0, ±1, ±2, ±3, • • • , (256)
and the energy values
m 2 h 2
Wi = £ (267)
In order to calculate the perturbation energy we shall need
integrals of the type
f*2ir r27r
H' m m> = ~M J *t**t'COS^iV= ^1 ««*'»>* cos **iV
J"2t f2ic
4ttJo
= for m' ?± m ± 1,
for m! = m ± 1.
M ,_ _, _ „ ^ W (258)
2
Using this result we see at once that the firstorder energy cor
rection is zero, for
W' m = EH' mm = 0. (259)
1 For a definition of /x see Equation 35.
2 This equation can be obtained as the approximate wave equation for a
system of two particles constrained by a potential function which restricts
the particles to a plane and keeps them a fixed distance apart by an argu
ment similar to that used in the discussion of the diatomic molecule men
tioned in the footnote to Section 35c.
178 PERTURBATION THEORY [VI25a
This problem is really a degenerate one, since W° depends only
on \m\ and not on the sign of m, so that there are two wave
functions for every energy level (other than the lowest). It is,
however, not necessary to consider this circumstance in evaluat
ing W' m and WZ because neither the first nor the secondorder
perturbation removes the degeneracy, and either the exponential
functions 256 or the corresponding sine and cosine functions are
satisfactory zerothorder wave functions.
The secondorder energy, as given by Equation 253, is
TJfftt _ JCT2 \"m,m— l) • F2 \"m,m+l) __ 4?T i/i A
Wm "Wl Wi_, + a Wl W° m+1 h\lm*  1)'
(2&10)
so that the total energy, to the second order, is
W , W o + XTr + v W „ , g^ + J£J*± _ . (2511)
It is interesting to point out the significance of this result
in connection with the effect of the electric field on the polariza
bility of the rotator. The polarizability a is the proportionality
factor between the induced dipole moment and the applied
field E. The energy of an induced dipole in a field is then
— %<xE 2 . From this and a comparison with Equation 2511 we
obtain the relation
which shows that a is positive for m = 0; the induced dipole
(whiqh in this case is due to the orienting effect of the field E on
the permanent dipole m of the rotator) is therefore in the direction
of the field E. For \m\ > 0, however, the opposite is true and
the field tends to orient the dipole in the reverse direction.
This is similar to the classicalmechanical result, which is
that a plane rotator with insufficient energy to make a complete
rotation in the field tends to be oriented parallel to the field
while a rotator with energy great enough to permit complete
rotation is speeded up when parallel and slowed down when
antiparallel to the field so that the resulting polarization is
opposed to the field. *
1 An interesting application of perturbation theory has been made to the
Stark effect of the hydrogen atom, the firstorder treatment having been
VI25a] SECONDORDER PERTURBATION THEORY 179
Problem 2&1. Carry out a treatment similar to the above treatment for
the rigid rotator in space, using the wave equation and wave functions found
in the footnote of Section 35c. Discuss the results from the viewpoint of
the last paragraph above. Compute the average contribution to the
polarizability of all the states with given / and with m = — /, — I f 1> • • • >
j /, assigning equal weights to the states in the averaging.
given independently by Schrodinger, Ann. d. Phys. 80, 437 (1926), and
P. S. Epstein, Phys. Rev. 28, 695 (1926), the second order by Epstein, loc.
cit., G. Wentzel, Z. /. Phys. 38, 518 (1926), and I.Waller, ibid. 38, 635 (1926),
and the third order by S. Doi, Y. Ishida, and S. Hiyama, Sci. Papers Tokyo
9, 1 (1928), and M. A. ElSherbini, Phil. Mag. 13, 24 (1932). See also
Sections 27a and 27e.
CHAPTER VII
THE VARIATION METHOD AND OTHER
APPROXIMATE METHODS
There are many problems of wave mechanics which cannot be
conveniently treated either by direct solution of the wave
equation or by the use of perturbation theory. The helium
atom, discussed in the next chapter, is such a system. No
direct method of solving the wave equation has been found
for this atom, and the application of perturbation theory is
unsatisfactory because the first approximation is not accurate
enough while the labor of calculating the higher approximations
is extremely great.
In many applications, however, there are methods available
which enable approximate values for the energy of certain of the
states of the system to be computed. In this chapter we shall
discuss some of these, paying particular attention to the variation
method, inasmuch as this method is especially applicable to the
lowest energy state of the system, which is the state of most
interest in chemical problems.
26. THE VARIATION METHOD
26a. The Variational Integral and Its Properties. — We shall
show 1 in this section that the integral
E = f<t>*H<t>dr (261)
is an upper limit to the energy Wo of the lowest state of a system.
In this equation, H is the complete Hamiltonian operator
H[ 5—. ry q ) for the system under discussfon (Sec. 12a) and <f>(q)
\Ziri o(j / f > ^ s . , ;
is any normalized function of the coordinates of the system
satisfying the auxiliary conditions of Section 9c for a satisfactory
wave function. The function <t> is otherwise completely unre
1 C. Eckart, Phys. Rev. 36, 878 (1930).
ISO
VH26al THE VARIATION METHOD 181
stricted; its choice may be quite arbitrary, but the more wisely
it is chosen the more closely will E approach the energy W .
If we used for our function <t>, called the variation function,
the true wave function \p of the lowest state, E would equal Wo)
that is,
E = MH+odr = Wo, (262)
since
/tyo = Wo4>o
If <f> is%ot equal to \po we may expand 4> in terms of the complete
set of normalized, orthogonal functions ^ , ^1, • • * , ^n, • • * ,
obtaining
<t> = %a n + n , with ^afan = 1. '■> (26^3)
n n
Substitution of this expansion in the integral for E leads to the
equation
E = ZX<*n>J+:H+ n ,dT = %a:a n W n , (26^)
n n' n
inasmuch as the functions ^ n satisfy the equations
Subtracting Wo, the lowest energy value, from both sides gives
E  Wo = J * ^*  TFo). (26^6)
Since TF n is greater than or equal to Wo for all values of n and the
coefficients a*a n are of course all positive oi* zero* the right side
of Equation 266 is positive or zero. We have therefore proved
that E is always an upper limit to Wo] that is,
E^Wo. " (267)
This theorem is the basis of the variation method for the
calculation of the approximate value of the lowest energy level
of a system. If we choose a number of variation functions
^1, <t>2, $*,'•• and calculate the values Ei, EQE h • • • cor
responding to them, then each of these values of H is greater
than the energy Wo, so that the lowest one is tlie nearest to Wo
Often the functions <t>i, fa, fa, • • • are only^distingui^^l by
having different values of some parameter. The process of
182 THE VARIATION METHOD lVH2&a
minimizing E with respect to this parameter may then be carried
out in order to obtain the best approximation to Wo which the
form of the trial function <t> will allow.
If good judgment has been exercised in choosing the trial
function <j> y especially if a number of parameters have been
introduced into <f> in such a manner as to allow its form to be
varied considerably, the value obtained for E may be very close
to the true energy TP . In the case of the helium atom, for
example, this method has been applied with great success, as is
discussed in the next chapter.
If E is equal to Wo then <t> is identical 1 with ^ (as can be
seen from Eq. 266), so that it is natural to assume that if E is
nearly equal to Wo the function 4> will approximate closely to
the true wave function ^ . The variation method is therefore
very frequently used to obtain approximate wave functions
as well as approximate energy values. From Equation 266
we see that the application of the variation method provides
us with that function <j) among those considered which approxi
mates most closely to fo according to the following criterion:
On expanding <j> — \f/ in terms of the correct wave functions ^ n ,
the quantity ^a n a n (W n — Wo) is minimized; that is, the sum
n
of the squares of the absolute values of the coefficients of the
wave functions for excited states with the weight factors W n — W
is minimized. For some purposes (as of course for the calcula
tion of the energy of the system) this is a good criterion to use ;
but for others the approximate wave function obtained in this
way might not be the most satisfactory one.
Eckart 2 has devised the following way of estimating how
closely a variation function approximates to the true solution ^ c
by using E and the experimental values of Wo and W%. A very
ieasonable criterion of the degree of approximation of <f> to ^o
(for real functions) is the smallness of the quantity
«/(* 4>o) 2 dr  /(**  2^oX^n + 4>l)dr = 2  2a ,
(268)
1 If the level Wo is degenerate, the equality of E and TT requires that
<t> be jpSitical with one of the wave functions corresponding to Wo.
* Reference on p. 180.
Vn26a] THE VARIATION METHOD 183
in which a is the coefficient of ^ in the expansion 263 of <t>.
From Equation 266 we can write
00 00
E  Wo = %a* n (W n  Wo) } %a* n (Wx  Wo)
n0 n=l
or
E  Wo > (W x  Wo)(l  aj).
Therefore if e 2 is small compared to €, we may combine this
equation and Equation 268, obtaining
^ E Wo , ^ 1 E W , „ n ,
€< f^¥ or l " < 2WT=w; (26 " 9)
Thus, from a knowledge of the correct energy values Wo and Wi
for the two lowest levels of the systems and the energy integral E
for a variation function <t>, we obtain an upper limit for the
deviation of a from unity, that is, of the contribution to </> of
wave functions other than i/'o.
The variation method has the great drawback of giving only an
upper limit to the energy, with no indication of how far from the
true energy that limit is. (In Section 26e we shall discuss a
closely related method, which is not, however, so easy to
apply, by means of which both an upper and a lower limit can be
obtained.) Nevertheless, it is very useful because there arise
many instances in which we have physical reasons for believing
that the wave function approximates to a certain form, and this
method enables these intuitions to be utilized in calculating a
better approximation to the energy than can be easily obtained
with the use of perturbation theory.
If we use for <j> the zerothorder approximation to the wave
function \pl discussed under perturbation theory, Chapter VI,
and consider H as equal to H° + H', this method gives for
E a value identical with the firstorder perturbation energy
Wl + W f o. If therefore we use for <f> a variation function con
taining parameters such that for certain values of the parameters
<t> reduces to ^§, the value we obtain for E is always at least as
good as that given by the firstorder perturbation treatment.
If 4> is set equal to the firstorder wave function, the energy value
E given by the variation method is the same, to the second
power in the parameter X, as the secondorder energy obtained
by the perturbation treatment.
184 THE VARIATION METHOD [VII26b
In case that it is not convenient to normalize <t>, the above
considerations retain their validity provided that E is given
by the expression
*  *fi%r ^
26b. An Example : The Normal State of the Helium Atom. —
In Section 236 we treated the normal state of the helium atom
with the use of firstorder perturbation theory. In this section
we shall show that the calculation of the energy can be greatly
increased in accuracy by considering the quantity Z which occurs
in the exponent (p = 2Zr/a ) of the zerothorder function given
in Equations 2334 and 2337 as a parameter Z' instead of as a
constant equal to the atomic number. The value of Z' is
determined by using the variation method with <f> given by
/ Z fz \ — Zri — Zrt
4 = ^fo = ( —  )e «»e ao , (2&11)
yra /
in which Z', the effective atomic number, is a variable parameter.
In this problem, the Hamiltonian operator is
H = 
87r 2 m,
 o <vl + vf)z«{i + i) + £
in which Z is the true atomic number. The factors <j>i and <£ 2
of <t> are hydrogenlike wave functions for nuclear charge Z'e,
so that 0i satisfies the equation
k2 "Vfri = ^~<t>i  Z'Wxfr (2612)
8ir*mo vrK1 . ri
(W H being equal to e 2 /2a ) y with a similar equation for <£ 2 . Using
these and the expression for H , we obtain
E = 2Z'*W a + (Z'  Z)e 2 </>*  +  Udr +
/*£+*>
i
<t>*—<t>dT. (2613)
T\2
The first integral on the right has the value
VII26b] THE VARIATION METHOD 185
j'3 f* °° C* C 2w \ 2Z ' ri 8Z' 3 e 2 C * 2Zri
o Jo Jo Jo n a\ Jo
2e 2 Z
==^ = 4Z'TF*. (2614)
The second integral of Equation 2613 is the same as that of
Equation 2338 if Z is replaced by Z'. It therefore has the
value
/
<t>*—<i>dT = ^Z'Wh. (2615)
r i2 4
Combining these results, we obtain for E the expression
E = { 2Z' 2 + %Z' + 4Z'(Z'  Z)\W H . (2&16)
Minimizing E with respect to Z' gives
i = = (4Z' + J + 8Z'  4z)jF*
or
2' = Z  % 6 , (2617)
which leads to
# = 2(Z  K*YWh. (2618)
As pointed out in Section 29c, this treatment cuts the error in
the energy of helium to onethird of the error in the firstorder
perturbation treatment. In the same section, more elaborate
variation functions are applied to this problem, with very
accurate results.
Problem 261. Calculate the energy of a normal hydrogen atom in &
uniform electric field of strength F along the z axis by the variation method,
and hence evaluate the polarizability a, such that the field energy is — J^aF 2 .
Use for the variation function the expression 1
1 The correct value of a for the normal hydrogen atom, given by the
secondorder perturbation theory (footnote at end of preceding chapter) is
a = % a * = 0.667 • 10~ 2 * cm 8 .
A value agreeing exactly with this has been obtained by the variation
method by H. R. Hasse*, Proc. Cambridge Phil Soc. 26, 642 (1930), using the
variation function ^i,(l + Az + Bzr). Hasse* also investigated the effects
of additional terms (cubic and quartic) in the series, finding them to be
negligible. The same result is given by the treatment of Section 27a.
186 THE VARIATION METHOD [VII26d
*i.(l + Az) y
minimizing the energy with respect to A, with neglect of powers of F higher
than F 2 .
a. = 4aJ = 0.59 • 10~ 24 cm 3 . Arts.
26c. Application of the Variation Method to Other States. —
The theorem E J> Wo, proved in Section 26a, may be extended
in special cases to states of the system other than the lowest
one. It is sometimes possible to choose <j> so that the first few
coefficients a 0f a if • • • of the expansion 263 are zero. If, for
example, a , ai, and a 2 are all zero, then by subtracting Wz
from both sides of Equation 264 we obtain
EW z = ^a n a n *{W n  W*) > 0, (2619)
n
since, although W  Wz, Wi  W*, and W 2  Wz are negative,
their coefficients are zero. In this case then we find the inequal
ity E J> Wz.
There are several cases in which such a situation may arise.
The simplest illustration is a onedimensional problem in which
the independent variable x goes from — oo to + °o and the
potential function V is an even function of x, so that
V(x) = V(+x).
The wave function belonging to the lowest level of such a system
is always an even function; i.e., \po( — x) = foix); while ^i is odd,
with \pi( — x) = —\l/\(x) (see Sec. 9c). If we therefore use for <j>
an even function, we can only say that E is greater than or equal
to Wo, but if 4> is an odd function, a will be zero (also all a n 's
with n even) and the relation E ^ Wi will hold. For such a
problem the variation method may be used to obtain the two
lowest energy levels.
The variation method may also be applied to the lowest state
of given resultant angular momentum and of given electronspin
multiplicity, as will be discussed in the next chapter (Sec. 29d).
Still another method of extending the variation method to levels
other than the lowest is given in the following section.
26d. Linear Variation Functions. 1 — A very convenient type of
variation function is one which is the sum of a number of linearly
x The generalized perturbation theory of Section 27a is closely related
to the treatment discussed here.
VII26d] THE VARIATION METHOD 187
independent functions xu X2, • • • , Xm with undetermined
coefficients c h c 2 , • • • , c m . In other words the variation
function <t> has the form
<t> = CiXl + C2X2 + " * * + CmXm, (2620)
in which ci, c 2 , • • • , c m are the parameters which are to be
determined to give the lowest value of E and therefore the best
approximation to Wo. It is assumed that the functions xu
X2, • • • , Xm satisfy the conditions of Section 9c. If we intro
duce the symbols
H nn > = fxnHxn'dr and A nn , = JxnXn'dr, (2621)
in which for simplicity we have assumed that <t> is real, then the
expression for E becomes
m m
/ . /* CnPn'tlnn'
J<t>H<t>dT n = l n'l (<>fr<l<X\
E ~ i*w ~ ^T^ ( '
nl n'l
or
E ^ ^CnCn'Ann* = j) ^C n C n 'ffnn'.
n n' n'
To find the values of c h c 2 , • • • , c m which make E a minimum,
we differentiate with respect to each c k :
* n n N w n
The condition for a minimum is that —  = for k = 1, 2, • • • ,
OCk
m, which leads to the set of equations
Xcn(H nk  AnkE) = 0, • * = 1, 2, • • • , m. (2623)
n
This is a set of m simultaneous homogeneous linear equations in
the m independent variables Ci, c 2 , • • • , c m . For this set of
188
THE VARIATION METHOD
[VII26d
equations to have a nontrivial solution it is necessary that the
determinant of the coefficients vanish (cf. Sec. 24); i.e., that
i/21
A21E
Hn ~ Ai2#
H 22 — A22E
H mi — AmlE H m 2 — &m2.E
H\ m
 Ai JE?
Him
 A 2 »^
tl mm
— A mm E
= 0.
(2624)
This equation is closely similar to the secular equation 2417 of
perturbation theory. It may be solved by numerical methods, 1
or otherwise, and the lowest root E = Eq is an upper limit to
Fia. 261.
Figure showing the interleaving of energy values for linear variation
functions with added terms.
the energy Wo Substitution of this value of E in Equations
2623 and solution of these equations for c 2 , c 6 , • • • , c m in
terms of ci (which can be used as a normalizing factor) gives the
variation function <j>o corresponding to E .
The other roots E h E 2 , • • • , E m \ of Equation 2624 are
upper limits for Wi, Wi, • • • , TT m i, respectively. 2 Further
more, it is possible to state how'these roots will be changed when
a new trial function <£' is used, containing one more function
Xm+l,
<t>' = CiXl + C2X2 + • * * + CmXm + C m+ iXm+l. (26~25)
In this case the roots E' 0y E[, E' 2)
the old ones E Qy E h E 2 , • •
• , 2^ will be separated by
E m as shown in Figure 261.
1 For a convenient numerical method see H. M. James and A. S. Coolidge,
J. Chem. Phys. 1, 825 (1933).
* J. K. L. MacDonald, Phys. Rev. 43, 830 (1933).
Vn26e] THE VARIATION METHOD 189
In other words, the relations E' ^ E , E[ ^ E lt etc., and
E ^ E[, Ei ^ E' 2 , etc., are satisfied.
This method has proved to be very useful in practice, as will
be illustrated by examples discussed in Chapters VIII and XII.
The application of the variation method to wave mechanics grew from the
work of Ritz, /. /. reine u. angew. Math. 135, 1 (1909), who considered the
solution of certain differential equations by discussing the equivalent
variation problem. It can be shown that a general normalized function fa
which satisfies the boundary conditions of Section 9c and which makes the
integral E — f<t>*HfadT a minimum relative to all variations in <£i is a
solution of the differential equation H\p = W^, E then being equal to the
corresponding characteristic energy value. A similar minimization of E
with respect to all variations in another general normalized function fa
with the added restriction that fa is orthogonal to <f>i leads to another solu
tion \p2 of the wave equation. By the continuation of this process of minimi
zation, all of the solutions can be found. Ritz proved that in certain cases
a rigorous solution can be obtained by applying a limiting process to the
integral /</>*#</>dr, in which <f> is represented as the sum of m functions of a
convenient set of normalized orthogonal functions ^i, ^2, • • • which satisfy
the boundary conditions, taken with arbitrary coefficients ci, C2, • • • , c».
For each value of m the coefficients c m are determined so that the integral
J<f>*H<f>dT is a minimum, keeping /</>*</>dr = 1. Ritz found that under
certain restrictions the sequence of functions converges to a true solution
of the wave equation and the sequence of values of the integral converges
to the corresponding true characteristic value. The approximate method
discussed in this section is very closely related to the Ritz method, differing
from it in that the functions yp are not necessarily members of a complete
orthogonal set and the limiting process is not carried out.
Problem 262. Using a variation function of the form <f> — A +• B cos
<p f C sin <p, obtain an upper limit to the lowest energy level of the plane
rotator in an electric field, for which the wave equation is
6H Sr 2 I
&i + ~~hT {W + * E cos ^ " °*
26e. A More General Variation Method. — A method has been
devised 1 which gives both an upper and a lower limit for an
energy level. If we represent by E and D the integrals
E = J<t>*H<t>dT and D = S(H4>)*(H4>)dr, (2626)
in which is a normalized trial variation function as before, then
we shall show that some energy level Wk satisfies the relation
E + VD  E 2 Z Wk} E  \/D  EK (26^27)
1 D. H. Wbinstbin, Proc. Nat, Acad, Sci. 20, 529 (1934); see also J. K. L.
MacDonald, Phys, Rev, 46, 828 (1934).
190 THE VARIATION METHOD [VII26e
To prove this we expand <t> as before (Eq. 263), so that
E = ^afanWn, D = ^aZanWl and ^a*a n = 1.
n n n
(2628)
From this we obtain the result
A = D  E* = %a£a n WZ  2£ga>„F„ + E^a*a n =
n n n
X^a n (W n E)\ (2629)
n
There will be some energy level Wk which lies at least as near
E as any other, i.e., for which
(Wt  Ey <c (w n  E)\
Therefore A is related to Wk — E by the inequality
A J> (Wk  E)^ata n
n
or
A ^ (W k  E)\ (2630)
There are now two possible cases,
Wk^E and W k < E.
In the first case we have
VA ^ W k  E, so that E + Va ^ W k ^ E;
and in the second case
VK >> E  W h and E > W k ^ E  VI.
From this we see that the condition in Equation 2627 applies
to both cases.
The application of this method to actual problems of the usual
type is more difficult than that of the simple variation method
because, in addition to the integral E, it is necessary to evaluate
D, which ordinarily is considerably more difficult than E.
It may be pointed out that by varying parameters in a function
in such a way as to make A a minimum the function <j> is made to
approach some correct wave function fa as closely as is permitted
by the form of <j>. This method consequently may be considered
as another type of variation method applicable to any state of a
system.
VII27a] OTHER APPROXIMATE METHODS 191
27. OTHER APPROXIMATE METHODS
There are a number of other methods which may be used to
obtain approximate wave functions and energy levels. Five of
these, a generalized perturbation method, the WentzelKramers
Brillouin method, the method of numerical integration, the
method of difference equations, and an approximate secondorder
perturbation treatment, are discussed in the following sections.
Another method which has been of some importance is based
on the polynomial method used in Section 11a to solve the
harmonic oscillator equation. Only under special circumstances
does the substitution of a series for \[/ lead to a twoterm recursion
formula for the coefficients, but a technique has been developed
which permits the computation of approximate energy levels for
lowlying states even when a threeterm recursion formula is
obtained. We shall discuss this method briefly in Section 42c.
27a. A Generalized Perturbation Theory. — A method of
approximate (and in some cases exact) solution of the wave
equation which has been found useful in many problems was
developed by Epstein 1 in 1926, immediately after the publication
of Schrodinger's first papers, and applied by him in the complete
treatment of the firstorder and secondorder Stark effects of the
hydrogen atom. The principal feature of the method is the
expansion of the wave function in terms of a complete set of
orthogonal functions which are not necessarily solutions of the
wave equation for any unperturbed system related to the system
under treatment, nor even necessarily orthogonal functions in
the same configuration space. Closely related discussions of
perturbation problems have since been given by a number of
authors, including Slater and Kirkwood 2 and Lennard Jones. 3
In the following paragraphs we shall first discuss the method in
general, then its application to perturbation problems and its
relation to ordinary perturbation theory (Chap. VI), and finally
as an illustration its application to the secondorder Stark effect
for the normal hydrogen atom.
In applying this method in the discussion of the wave equation
H+(x) = W+(x), (271)
1 P. S. Epstein, Phys. Rev. 28, 695 (1926).
* J. C. Slater and J. G. Kirkwood, Phys. Rev. 37, 682 (1931).
8 J. E. LennardJones, Proc. Roy. Soc. A 129, 598 (1930).
192 THE VARIATION METHOD [VII27a
in which x is used to represent all of the independent variables
for the system, we express \l/(x) in terms of certain functions
Fn(x), writing
*(*) = %AnF n (x). (272)
n
The functions F n (x) are conveniently taken as the members of a
complete set of orthogonal functions of the variables a;; it is not
necessary, however, that they be orthogonal in the same con
figuration space as that for the system under discussion. Instead,
we assume that they satisfy the normalization and orthogonality
conditions
JF*(x)F n (x)p(x)dx = 8 mn \
with
flform = nA ( 27 ~ 3 )
for w^wj
i 1
in which p{x)dx may be different from the volume element dr
corresponding to the wave equation 271. p{x) is called the
weight factor 1 for the functions F n (x). On substituting the
expression 272 in Equation 271, we obtain
%A n (H  W)F n (x) = 0, (274)
n
which on multiplication by F*(x)p{x)dx and integration becomes
%A n (H mn ~ Wbmn) = 0, TTl = 1, 2, • • • , (275)
n
in which
#mn = fFZ(x)HF n (x)p(x)dx. (276)
1 In case that the functions F n (x) satisfy the differential equation
£{p(*)^}  «<*>* + *>(*)* m °>
in which X is the characteristicvalue parameter, they are known to form
a complete set of functions which are orthogonal with respect to the weight
factor p(x) . For a discussion of this point and other properties of differential
equations of the SturmLiouviile type see, for example, R. Courant and
D. Hilbert, "Methoden der mathematischen Physik," Julius Springer,
Berlin, 1031.
VH27al
OTHER APPROXIMATE METHODS
193
For an arbitrary choice of the functions F n (x) Equation 275
represents an infinite number of equations in an infinite number
of unknown coefficients A n . Under these circumstances ques
tions of convergence arise which are not always easily answered.
In special cases, however, only a finite number of functions
F n (x) will be needed to represent a given function ^(x) ; in these
cases we know that the set of simultaneous homogeneous linear
equations 275 has a nontrivial solution only when the deter
minant of the coefficients of the i n 's vanishes; that is, when the
condition
Hn H22 ~~ W Hiz
H31 H 32 i/33 — W
= (277)
is satisfied. We shall assume that in the infinite case the mathe
matical questions of convergence have been settled, and that
Equation 277, involving a convergent infinite determinant, is
applicable.
Our problem is now in principle solved: We need only to eval
uate the roots of Equation 277 to obtain the allowed energy
values for the original wave equation, and substitute them in
the set of equations 275 to evaluate the coefficients A n and
obtain the wave functions.
The relation of this treatment to the perturbation theory of
Chapter VI can be seen from the following arguments. If the
functions F n (x) were the true solutions ^»(x) of the wave equation
271, the determinantal equation 277 would have the form
WiW
W 2  W
Wz  W
= 0, (278)
with roots W = W\ f W = W 2 , etc. Now, if the functions
F n (x) closely approximate the true solutions rp n (x), the non
diagonal terms in Equation 277 will be small, and as an approxi
mation we can neglect them. This gives
TTi = ffn,
W 2 = #22,
Wz = #33>
etc.,
(279)
194
THE VARIATION METHOD
[Vn27a
which corresponds to ordinary firstorder perturbation theory,
inasmuch as, if # can be written as H° + #', with
H°F n (x) = W°F n (x),
then W n = H nn has the value W n = W° n + JFZ(x)H'F n (x)p(x)dx,
which is identical with the result of ordinary firstorder per
turbation theory of Section 23 when p{x)dx = dr. Equation
279 is more general than the corresponding equation of first
order perturbation theory, since the functions F n (x) need not
correspond to any unperturbed system. On the other hand,
it may not be so reliable, in case that a poor choice of functions
F n (x) is made; the first step of ordinary perturbation theory is
essentially a procedure for finding suitable zerothorder functions.
It may happen that some of the nondiagonal terms are large
and others small; in this case neglect of the small terms leads to
an equation such as
#n  W #12
#21 #22  W
#33  W
#44  W
which can be factored into the equations
= o,
ffll
w
H 2
Ha
#22  W
= 0,
H 33  W
= o,
H it  W
= 0,
etc.
(2710)
It is seen that this treatment is analogous to the firstorder
perturbation treatment for degenerate states as given in Section
24. The more general treatment now under discussion is espe
cially valuable in case that the unperturbed levels are not exactly
equal, that is, in case of approximate degeneracy.
A second approximation to the solution of Equation 277
can be made in the following manner. Suppose that we are
interested in the second energy level, for which the value #22
is found for the energy as a first approximation. We introduce
this expression for W everywhere except in the term #22 — W,
Vn27a]
OTHER APPROXIMATE METHODS
195
and neglect nondiagonal terms except H 2n and //» 2 , thus obtain
ing the equation
Hu — H22 H12
H21 H22 ~ W H23 H24
il 32 2/33 ~" H22
H42 Ha — H22
= 0.
(2711)
On multiplying out the determinant, we convert this equation
into the form
(#22  >WKH n  H 2i )(H 33  HnHHu
— i/i2#2l(//33 ~ H22) {Hu — H22) ' ' '
— Hz2H2z(H\\ — H22)(Hu — H22) • • * —
H22)
with the solution
W
H » ~ 2 7/~
H21H12
H 2
0,
(2712)
in which the prime indicates that the term with I = 2 is omitted.
This is analogous to (and more general than) the secondorder
perturbation treatment of Section 25; Equation 2712 becomes
identical with Equation 253 when Hu is replaced by Wi and
H %1 by H' kl .
Higher approximations can be carried out by obvious exten
sions of this method. If Equation 277 can be factored into
equations of finite degree, they can often be solved accurately by
algebraic or numerical methods.
Let us now consider a simple example, 1 the secondorder
Stark effect of the normal hydrogen atom, using essentially the
method of Epstein (mentioned above). This will also enable
us to introduce and discuss a useful set of orthogonal functions.
The wave equation for a hydrogen atom in an electric field
can be written as
8tt 2 m
V 2^ _ L.^ + eFz ^ = w ^ f
(2713)
in which eFz represents the interaction with an electric field of
strength F along the z axis. In order to discuss this equation we
shall make use of certain functions /^x M (£, #, <p), defined in terms
1 The study of this example can be omitted by the reader if desired.
196 THE VARIATION METHOD [VH27a
of the associated Laguerre and Legendre functions (Sees. 19
and 20) as
*Wtt, *, <p) = A*(QO*(mM, (2714)
in which
A *<«> = [ferw'T^'^"'' (27_15)
LJ+j^U) being an associated Laguerre polynomial as defined in
Section 206. The functions 6\ M (#) and $ M (^) are identical with
the functions 6/m(#) and $ m (<p) of Equations 212 and 213
except for the replacement of I and m by X and /z. It is found
by the use of relations given in Sections 19 and 20 that F v \^ } #, <p)
satisfies the differential equation
W 4 ? §F (v _ 1 \ F 1 dW
dp f a? \f V £ 2 sin 2 a^ 2
+ £ 2 sin 1
The functions are normalized and mutually orthogonal with
weight factor £ , satisfying the relations
(v = v'
JTJCJC >WWM; sin MM, = 1 for X « X'(
= otherwise.
If we identify £ with 2Zr/n'a , where a = h 2 /^K 2 ixe 2 7 then
the functions F„\ M become identical with the hydrogenatom wave
functions \pnim for the value n = n' of the principal quantum
number n, but not for other values of n; the functions F„x M all
contain the same exponential function of r, whereas the hydrogen
atom wave functions for different values of n contain different
exponential functions of r.
For the problem at hand we place n' equal to 1 and Z equal to 1,
writing
2r _ h 2
The functions F„x M then satisfy the equation
V'F, Xm + Q  0F,x M  (y = 1} * W (2719)
*  > «o  335 (2718)
Vn27a] OTHER APPROXIMATE METHODS
Now let us write our wave equation 2713 as
vv +
in which
(H>
A ~ 4e'
A£co8&\f/ = fi\py
=
Wa
' 2e 2
1
i'J
197
(2720)
(2721)
and the operation V 2 refers to the coordinate £ rather than r,
£ being given by Equation 2718. To obtain an approximate
solution of this equation in terms of the functions F^ we shall
set up the secular equation in the form corresponding to. second
order perturbation theory for the normal state, as given in
Equation 2711; we thus obtain the equation
"21 H22
H*i Hm
= 0, (2722)
in which
Hi, = jj Jf? (v 2 +t  i  M«*
*
•)
F£*d( sin UUip, (2723)
i and j being used to represent the three indices v, X, /x. The
factor 2 before p arises from the fact that the functions F„\p
are not normalized to unity with respect to the volume element
£ 2 d£ sin $d#d<p.
It is found on setting up the secular equation 2722 that only
the three functions F100, ^210, and F310 need be considered, inasmuch
as the equation factors into a term involving these three func
tions only (to the degree of approximation considered) and terms
involving other functions. The equations
and
£ 2 cos &F 100 = 4\/2F 21 o  2\Z2Fz
(2724)
€**,  ~ {("  *)(» + X + 1)}^+i.xm + 2vF,xm 
{ (v + \)(v  X  1) } W,^, (2725)
198 THE VARIATION METHOD [VII27b
together with Equation 2717, enable us to write as the secular
equation for these three functions
2/3 4\/2A 2V2A
4V2A 1 =0. (2726)
2y/2A 2
The root of this is easily found to be fi = ISA 2 , which corresponds
to
e 2 9
or
W" = W W° = V^W 2 ' (2727)
This corresponds to the value
a = y 2 a\ = 0.677 • 10" cm 3
for the polarizability of the normal hydrogen atom.
Problem 271. Derive the formulas 2724 and 2725.
Problem 272. Discuss the firstorder and* secondorder Stark effects
for the states n = 2 of the hydrogen atom by the use of the functions F„xm
Note that in this case the term in A can be neglected in calculating
tl v'\' n' ,v' '\' V"
unless v or v" is equal to 2, and that the secular equation can be factored
into terms for p. = \l, n =0, and \x — — 1, respectively.
27b. The WentzelKramersBrillouin Method. — For large
values of the quantum numbers or of the masses of the particles
in the system the quantum mechanics gives results closely similar
to classical mechanics, as we have seen in several illustrations.
For intermediate cases it is found that the old quantum theory
often gives good results. It is therefore pleasing that there
has been obtained 1 an approximate method of solution of the
wave equation based on an expansion the first term of which
leads to the classical result, the second term to the oldquantum
theory result, and the higher terms to corrections which bring
in the effects characteristic of the new mechanics. This method
is usually called the WentzelKramersBrillouin method. In
our discussion we shall merely outline the principles involved
in it.
1 G. Wentzel, Z. /. Phys. 38, 518 (1926); H. A. Kramers, Z.f. Phys. 39,
828 (1926); L. Brillouin, J. de phys. 7, 353 (1926); J. L. Dunham, Phyg.
Rev. 41, 713 (1932).
VH27b] OTHER APPROXIMATE METHODS 199
For a onedimensional problem, the wave equation is
g + ^ar™ = o, oo<*< + oo.
If we make the substitution
+ = ffi vd \ (2728)
we obtain, as the equation for y,
A g = 2m (W V)y* = p> y\ (2729)
in which p = ±\/2m(W — V) is the classical expression for
the momentum of the particle. We may now expand y in powers
of h/2iri } considering it as a function of h, obtaining
y  f • + £* + (as)* + ' ' • • (27 " 30)
Substituting this expansion in Equation 2729 and equating the
coefficients of the successive powers of h/2iri to zero, we obtain
the equations
Vo
= p = ±V2m(W  V), (2731)
yi = iio == ~~k = 4(tt  vy  (27 ~ 32)
y2 = ~^ 2 f5F' 2 +4F"(lf  7)}(2ro)H(TP  F)* (2733)
dF rf 2 F
in which F ; = ^ and V" = ^4
aa; ax 2
The first two terms when substituted in Equation 2728 lead
to the expression
1 2«i
* S W  V) *e a" J^™* (2734)
as an approximate wave function, since
J »i& = J ^zrp^ = + ij wzrv = \ lo s W ^ F )
so that
The probability distribution function to this degree of approxima
tion is therefore
*** = NKW  F)H = const, i (2735)
200 THE VARIATION METHOD lVH27b
agreeing with the classical result, since p is proportional to the
velocity and the probability of finding a particle in a range dx is
inversely proportional to its velocity in the interval dx.
The approximation given in Equation 2734 is obviously not
valid near the classical turning points of the motion, at which
W = V. This is related to the fact that the expansion in
Equation 2730 is not a convergent series but is only an asymp
totic representation of y, accurate at a distance from the points
at which W = V.
So far nothing corresponding to quantization has appeared.
This occurs only when an attempt is made to extend the wave
function beyond the points W = V into the region with W less
than V. It is found 1 that it is not possible to construct an
approximate solution in this region satisfying the conditions
of Section 9c and fitting smoothly on to the function of Equation
2734, which holds for the classically allowed region, unless W
is restricted to certain discrete values. The condition imposed
on W corresponds to the restriction
fydx = nh, n = 0, 1, 2, 3, • • • , (2736)
in which the integral is a phase integral of the type discussed in
Section 56. If we insert the first term of the series for y, y = p,
we obtain the oldquantumtheory condition (Sec. 56)
fpdx = nh, n = 0, 1, 2, 3 • • • . (2737)
For systems of the type under discussion, the second term
h
5i*
introduces half quantum numbers; i.e., with y = y + 75.2/1,
(bydx = Spdx + ~—.<£yidx = (Lpdx — ^ = nh,
so that
fpdx = (n + V 2 )h (2738)
to the second approximation. (The evaluation of integrals such
as jfyidx is best carried out by using the methods of complex
variable theory, which we shall not discuss here. 2 )
This method has been applied to a number of problems and
is a convenient one for many types of application. Its main
1 Even in its simplest form the discussion of this point is too involved to
be given in detail here.
* J. L. Dunham, Phy* Rev. 41, 713 (1932).
VH27C] OTHER APPROXIMATE METHODS 201
drawback is the necessity of a knowledge of contour integration,
but the labor involved in obtaining the energy levels is often
considerably less than other methods require.
27c. Numerical Integration. — There exist welldeveloped meth
ods 1 for the numerical integration of total differential equations
which can be applied quite rapidly by a practiced investigator.
The problem is not quite so simple when it is desired to find
characteristic values such as the energy levels of the wave equa
tion, but the method is still practicable.
Hartree, 2 whose method of treating complex atoms is dis
cussed in Chapter IX, utilizes the following procedure. For
some assumed value of W, the wave equation is integrated
numerically, starting with a trial function which satisfies the
boundary conditions at one end of the range of the independent
variable x and carrying the solution into the middle of the range
Another solution is then computed for this same value of W,
starting with a function which satisfies the boundary conditions
at the other end of the range of x. For arbitrary values of W
these two solutions will not in general join smoothly when they
meet for some intermediate value of x. W is then changed
by a small amount and the process repeated. After several
trials a value of IF is found such that the righthand and left
hand solutions join together smoothly (i.e., with the same
slope), giving a single wave function satisfying all the boundary
conditions.
This method is a quantitative application of the qualitative
ideas discussed in Section 9c. The process of numerical integra
tion consists of starting with a given value and slope for ^ at a
point A and then calculating the value of ^ at a nearby point B
by the use of values of the slope and curvature 75 at A, the latter
being obtained from the wave equation.
This procedure is useful only for total differential equations in
one independent variable, but there are many problems involving
several independent variables which can be separated into total
1 E. P. Adams, "Smithsonian Mathematical Formulae," Chap. X, The
Smithsonian Institution, Washington, 1922; E. T. Whittaker and G.
Robinson, "Calculus of Observations," Chap. XIV, Blackie and Son., Ltd.,
London, 1929.
1 D. R. Hartree, Proc. Cambridge Phil. Soc. 24, 105 (1928); Mem.
Manchester Phil. Soc. 77, 91 (19321933).
202
THE VARIATION METHOD
[VII27d
differential equations to which this method may then be applied.
Hartree's method of treating complicated atoms (Sec. 32) and
Bureau's calculation of the energy of HJ (Sec. 42c) are illustra
tions Oa the use of numerical integration.
27d. Approximation by the Use of Difference Equations.— The
wave equation
g + k 2 (W  V)+ = o,
_8x 2 m
10 " h 2 '
(2739)
may be approximated by a set of difference equations, 1
±(fc!  2h + * m ) + k 2 {W  7(x,)}*< = 0, (2740)
or
5)m, = w^„
(2741)
in which ^i, ^ 2 ,
, *<,
function ^ at the points Zi, x 2 ,
are numbers, the values of the
■••,£»,••• uniformly spaced
Fiq. 271. — The approximation to a wave function ¥ by segments of straight lines.
along the x axis with a separation x» — s*i = a. To prove this
we consider the approximation to \p formed by the polygon of
straight lines joining the points (x\> ^i), (z 2 , ^2), • • • >
(s», ^*), • • • of Figure 271. The slope of ^ at the point
halfway between Xii and Xi is approximately equal to the slope
of the straight line connecting x^i and x i} which is (^ — ^;_i)/a.
The second derivative of ^ at x = Xi is likewise approximated by
1/a times the change in slope from (xi + Xii)/2 to (a;* + Xi+i)/2;
that is,
dx 2 a 2
(2742)
1 R. G. D. Richardson, Trans. Am. Math. Soc. 18, 489 (1917); R. Courant,
\C ESannivnTriTra anH "FT T.mwv Mnthrmntijirh* Ainnnlein Iftft 39 (1 Q9ft"\
VH27d] OTHER APPROXIMATE METHODS 203
The differential equation 2739 is the relation between the
curvature ^ata point and the function k 2 (W — V)^ at that
point, so that we may approximate to the differential equation
by the set of equations 2740, there being one such equation for
each point x». The more closely we space the points x if the more
accurately do Equations 2740 correspond to Equation 2739.
Just as the lowest energy W of the differential equation can be
obtained by minimizing the energy integral E = f<t>*H<t>dT with
respect to the function <£, keeping /</>*</>dr = 1, so the lowest
value of W giving a solution of Equations 2740 may be obtained
by minimizing the quadratic form
2^bi } (t>i<t>j
E = 2— , (2743)
X
in which to, to, • • • , 4>i, • • • are numbers which are varied
until E is a minimum. (Just as <t> must obey the boundary
conditions of Section 9c, so the numbers # t must likewise approxi
mate a curve which is a satisfactory wave function.)
A convenient method 1 has been devised for carrying out this
minimization. A set of trial values of <£» is chosen and the
value of E is calculated from them. The true solutions \p i}
to which the values of to will converge as we carry out the
variation, satisfy Equations 2740. Transposing one of these
gives
*' = 2  aW[W  Viz*)}' (27_44)
If the (fc's we choose are near enough to the true values ^», then
it can be shown 1 that, by putting toi and </> 1+ i in place of \pii
and yj/i+i and E in place of W in Equation 2744, the resulting
expression gives an improved value ^ for <j> i} namely,
In this way a new set <t>[, to, • • • , <£•, • • • can be built up
from the initial set fa, to,  •  , <t>i, •  • t ^e new set giving a
1 G. E. Kimball and G. H. Shortley, Phys. Rev. 45, 815 (1934).
204 THE VARIATION METHOD [VII27e
lower and therefore a better value of E than the first set. This
process may then be repeated until the best set <£» is obtained
and the best value of E.
This procedure may be modified by the use of unequal intervals,
and it can be applied to problems in two or more dimensions,
but the difficulty becomes much greater in the case of two
dimensions.
Problem 273. Using the method of difference equations with an
interval a = }4, obtain an upper limit to the lowest energy Wo and an
approximation to ^ for the harmonic oscillator, with wave equation
^ + (X  x*)+ = (see Eq. 111).
27e. An Approximate Secondorder Perturbation Treatment.
The equation for the secondorder perturbation energy (Eq. 253)
is
w '»'  2 'w~ff + H "» ( 27  46)
i
with
H' u = M*H'+fdT
and
The sum may be rearranged in such a manner as to permit an
approximate value to be easily found. On multiplying by
W + W ' becomes
Wl^ kl lk "*" ^J Wl (W° k  wtf
I I
Now we can replace 1 ^'H' kl H' lk by (H' 2 ) kk  {H f kk )\ obtaining
i
1 To prove this, we note that H'ypl = £jH' lk \l/\ (as is easily verified by
I
multiplication by \p t * and integration). Hence
(#")**  f+l'H'VldT  f+l*H'H'+ldr = fr k *H'(%H' lk +tydT
The sum JJ « #m#j* differs from this only by the term with I  A;, (H'h)*.
VII27e] OTHER APPROXIMATE METHODS 205
for W' k ' the expression
W"  (#' ')** (H'kk) 2 , Tin , N? Wj #;»#!* . (<Y7_A7\
Wk " "W! WT + "** + ^j W2 (W2  Wf) { }
in which
(ff'% = M*H'V k dr. (2748)
This expression is of course as difficult to evaluate as the
original expression 2746. However, it may be that the sum is
small compared with the other terms. For example, if k repre
sents the normal state of the system, and the origin for energy
measurements is such that Wl is negative, the terms in the sum
will be negative, for W® negative, and positive for W? positive,
and there may be considerable cancellation. It must be empha
sized that the individual terms in this expression are dependent
on the origin chosen for the measurement of energy (the necessity
for an arbitrary choice of this origin being the main defect of the
approximate treatment we are describing). If this origin were
to be suitably chosen, this sum could be made to vanish, the
secondorder perturbation energy then being given by integrals
involving only one unperturbed wave function, that for the
state under consideration. The approximate treatment consists
in omitting the sum.
As an example let us take the now familiar problem of the
polarizability of the normal hydrogen atom, with H' = eFz.
We know that H' uu vanishes. The integral (H' 2 ) u ,u is equal
to e 2 F 2 (z 2 ) u , U) and, inasmuch as r 2 = x 2 + y 2 + z 2 and the
wave function for the normal state is spherically symmetrical,
the value of (z 2 )i,,u is just onethird that of (r 2 ) UfU , given in
Section 21c as 3ag. Thus we obtain
e 2 F 2 al
\n" = °
Wl
If we use the value — e 2 /2a for Wl, (taking the ionized atom at
zero energy), we obtain
W" = 2F 2 a 3 ,
which corresponds to the value a = 4aJ for the polarizability.
This is only II per cent less than the true value (Sec. 27a),
being just equal to the value given by the simple treatment of
Problem 261.
206 THE VARIATION METHOD [Vn27e
It is interesting to note that if, in discussing the normal state
of a system, we take as the zero of energy the first unperturbed
excited level, then the sum is necessarily positive and the approxi
mate treatment gives a lower limit to W". In the problem of the
normal hydrogen atom this leads to
giving for a the upper limit x %al f which is 18 per cent larger
than the correct value %a%. Inasmuch as the value a = 4a§ given
by the variation method is a lower limit, these two very simple
calculations fix a to within a few per cent.
It was pointed out by Lennard Jones 1 that this approximate
treatment of W' k ' corresponds to taking as the firstorder per
turbed wave function the approximate expression (not normalized)
** = «(1 + AH' + • • • ) (2749)
in which A = 1/Wl
This suggests that, when practicable, it may be desirable to
introduce the perturbation function in the variation function
in this way in carrying out a variation treatment. Examples of
calculations in which this is done are given in Sections 29#
and 47.
1 J. E. Lennard Jones, Proc. Roy. Soc. A129, 598 (1930).
CHAPTER VIII
THE SPINNING ELECTRON AND THE PAULI
EXCLUSION PRINCIPLE, WITH A DISCUSSION OF THE
HELIUM ATOM
28. THE SPINNING ELECTRON 1
The expression obtained in Chapter V for the energy levels of
the hydrogen atom does not account completely for the lines
observed in the hydrogen spectrum, inasmuch as many of the
lines show a splitting into several components, corresponding
to a fine structure of the energy levels not indicated by the simple
theory. An apparently satisfactory quantitative explanation
of this fine structure was given in 1916 by the brilliant* work of
Sommerfeld, 2 who showed that the consideration of the rela
tivists change in mass of the electron caused the energy levels
given by the old quantum theory to depend to some extent on
the azimuthal quantum number k as well as on the total quantum
number n, the splitting being just that observed experimentally
not only for hydrogen and ionized helium but also for xray
lines of heavy atoms. This explanation was accepted for
several years. Shortly before the development of the quantum
mechanics, however, it became evident that there were trouble
some features connected with it, relating in particular to the
spectra of alkalilike atoms. A neutral alkali atom consists in
its normal state of an alkali ion of particularly simple electronic
structure (a completed outer group of two or eight electrons)
and one valence electron. The interaction of the valence electron
and the ion is such as to cause the energy of the atom in various
quantum states to depend largely on the azimuthal quantum
number for the valence electron as well as on its total quantum
number, even neglecting the small relativistic effect, which ip
negligible compared with the electronion interaction. How
1 For a more detailed treatment of this subject see L. Pauling and S.
Goudsmit, "The Structure of Line Spectra," Chap. IV.
2 A. Sommerfeld, Ann. d. Phys. 51, 1 (1916).
207
208 THE SPINNING ELECTRON [Vm28
ever, the levels corresponding to given values of these two
quantum numbers were found to be often split into two levels,
and it was found that the separations of these doublet levels are
formally representable by the Sommerfeld relativistic equation.
Millikan and Bowen 1 and Land6, 2 who made this discovery,
pointed out that it was impossible to accept the relativistic
mechanism in this case, inasmuch as the azimuthal quantum
number is the same for the two components of a doublet level,
and they posed the question as to the nature of the phenomenon
involved.
The answer was soon given by Uhlenbeck and Goudsmit, 3
who showed that the difficulties were removed by attributing to
the electron the new properties of angular momentum and
magnetic moment, such as would be associated with the spinning
motion of an electrically charged body about an axis through
it. The magnitude of the total angular momentum of the
electron is y/s(s + l)^> in which s, the electronspin quantum
number, is required by the experimental data to have the value
3^. The component of angular momentum which the electron
spin can possess along any prescribed axis is either +rr or
sri that is, it is given by the expression m,~, in which the
quantum number m, can assume only the values +}4 an d —}i*
To account for the observed finestructure splitting and Zeeman
effects it is found that the magnetic moment associated with
the electron spin is to be obtained from its angular momentum
by multiplication not by the factor e/2woc, as in the case of
orbital magnetic moment (Sec. 21d), but by twice this factor,
the extra factor 2 being called the* Landi g factor for electron spin.
In consequence the total magnetic moment of the electron spin
1 R. A. Millikan and I. S. Bowen, Phys. Rev. 24, 223 (1924).
* A. Landb, Z. /. Phys. 25, 46 (1924).
* G. E. Uhlenbeck and S. Goudsmit, Naiurwis&enschaflen 13, 953 (1925);
Nature 117, 264 (1926). The electron spin was independently postulated by
R. Bichowsky and H. C. Urey, Proc. Nat. Acad. Set. 12, 80 (1926) (in whose
calculations there was a numerical error) and had been previously suggested
on the basis of unconvincing evidence by several people.
Vm281 THE SPINNING ELECTRON 209
is 2~ o~\l •  or a/3 Bohr magnetons, and the component along
a prescribed axis is either +1 or — 1 Bohr magneton.
It was shown by Uhlenbeck and Goudsmit and others 1 that the
theory of the spinning electron resolves the previous difficulties,
and the electron spin is now accepted as a property of the electron
almost as well founded as its charge or mass. The doublet
splitting for alkalilike atoms is due purely to the magnetic
interaction of the spin of the electron and its orbital motion.
The fine structure of the levels of hydrogenlike atoms is due to
a particular combination of spin and relativity effects, resulting
in an equation identical with Sommerfeld's original relativistic
equation. The anomalous Zeeman effect shown by most atoms
(the very complicated splitting of spectral lines by a magnetic
field) results from the interaction of the field with both the
orbital and the spin magnetic moments of the electrons, the
complexity of the effect resulting from the anomalous value 2
for the g factor for electron spin. 2
The theory of the spinning electron has been put on a particu
larly satisfactory basis by the work of Dirac. In striving to
construct a quantum mechanics compatible with the require
ments of the theory of relativity, Dirac 3 was led to a set of
equations representing a oneelectron system which is very
different in form from the nonrelativistic quantummechanical
equations which we are discussing. On solving these, he found
that the spin of the electron and the anomalous g factor 2 were
obtained automatically, without the necessity of a separate
postulate. The equations led to the complete expression for
the energy levels for a hydrogenlike atom, with fine structure,
and even to the foreshadowing of the positive electron or positron,
discovered four years later by Anderson.
So far the Dirac theory has not been extended to systems
containing several electrons. Various methods of introducing
1 W. Pauli, Z. f. Phys. 36, 336 (1926); W. Heisenberg and P. Jordan,
Z. /. Phys. 37, 266 (1926); W. Gordon, Z. f. Phys. 48, 11 (1928); C. G.
Darwin, Proc. Roy. Soc. A 118, 654 (1928); A. Sommerfeld and A. Unsold,
Z. f. Phys. 36, 259; 38, 237 (1926).
'For a fuller discussion see Pauling and Goudsmit, "The Structure of
line Spectra," Sees. 17 and 27.
* P. A. M. Dirac, Proc. Roy. Soc. A117, 610; A118, 351 (1928).
210 THE SPINNING ELECTRON [VHI29a
the spin in nonrelativistic quantum mechanics have been
devised. Of these we shall describe and use only the simplest
one, which is satisfactory so long as magnetic interactions
are neglected, as can be done in treating most chemical and
physical problems. This method consists in introducing a
spin variable w, representing the orientation of the electron,
and two spin wave functions, a(o>) and /3(w), the former cor
responding to the value +J^ for the spincomponent quantum
number m a (that is, to a component of spin angular momentum
along a prescribed axis in space of +} / 2h/2r) and the latter to
the value — \^ f or m « The two wave functions are normalized
and mutually orthogonal, so that they satisfy the equations
/a 2 (co)da> = 1, )
J*0 2 (co)dco = 1, \ (281)
Ja(co)j8(co)doj = 0.)
A wave function representing a oneelectron system is then
a function of four coordinates, three positional coordinates
such as x, y, and z, and the spin coordinate w. Thus we write
$(%, V, 2)<*(co) and \f/(x, y, z)/3(co) as the two wave functions cor
responding to the positional wave function \[/(x, y, z), which is a
solution of the Schrodinger wave equation. The introduction
of the spin wave functions for systems containing several electrons
will be discussed later.
Various other simplified methods of treating electron spin have
been developed, such as those of Pauli, 1 Darwin, 2 and Dirac. 3
These are especially useful in the approximate evaluation of
interaction energies involving electron spins in systems containing
more than one electron.
29. THE HELIUM ATOM. THE PAULI EXCLUSION PRINCIPLE
29a. The Configurations 1525 and 152^. — In Section 236 we
applied the firstorder perturbation theory to the normal helium
atom. Let us now similarly discuss the first excited states of
this atom, 4 arising from the unperturbed level for which one
1 W. Pauli, Z. f. Phya. 43, 601 (1927).
* C. G. Darwin, Proc. Roy. Soc. A116, 227 (1927).
8 P. A. M. Dirac, Proc. Roy. Soc. A123, 714 (1929).
4 This was first done by W. Heisenberg, Z. f. Phys. 39, 499 (1926).
Vm29a]
THE HELIUM ATOM
211
electron has the total quantum number n = 1 and the other
n = 2. It was shown that, if the interelectronic interaction
term e 2 /ri 2 be considered as a perturbation, the solutions of the
unperturbed wave equation are the products of two hydrogen
like wave functions
in which the symbol (1) represents the coordinates (r u th, <pi)
of the first electron, and (2) those of the second electron. The
corresponding zerothorder energy is
W° nn = ^Rhc( 2 + \
We shall ignore the contribution of electron spin to the wave
function until the next section.
The first excited level, with the energy W Q = —5Rhc, is that
for m = 1, n 2 = 2 and rii = 2, n 2 = 1. This is eightfold
degenerate, the eight corresponding zerothorder wave functions
being
i«(D
2»(1)
1«(1)
2p,(l)
1.(1)
2p„(l)
1.(1)
2p.(l)
2s(2),
1«(2),
2p,(2)J
1«(2),
2p»(2),/
1«(2),
2p.(2),]
18(2),
(291)
in which we have chosen to use the real y functions and have
represented ^ioo(l) by ls(l), and so on.
On setting up the secular equation, it is found to have the
form
j,  w
K.
K.
/.  w
J p 
w
Kp
K P
Jp
w
Jp  w
Kp
K P
Jp
 w
Jp
 w
Kp
Kp
Jp
 W
(2&~2)
212 THE SPINNING ELECTRON [VIII29a
Here the symbols J», K a , J vy and K v represent the perturbation
integrals
(293)
1«(1) Sto(2) — 1«(1) 25(2) dudTz,
7*12
1«(1) 2s(2) — 2s(l) 1«(2) dndr h
1*(1) 2p,(2) ^ 1«(1) 2p x (2) dndrt, 
#p = f fl*(l) 2px(2) — 2p,(l) 1*(2) dndr,.
J p and 2£ p also represent the integrals obtained by replacing
2p x by 2p„ or 2p*, inasmuch as these three functions differ from
one another only with regard to orientation in space. The
integrals J, and J p are usually called Coulomb integrals; J,,
for example, may be considered to represent the average Coulomb
interaction energy of two electrons whose probability distribution
functions are (ls(l)} 2 and {2s(2)} 2 . The integrals K 9 and K p
are usually called resonance integrals (Sec. 41), and sometimes
exchange integrals or interchange integrals, since the two wave
functions involved differ from one another in the interchange
of the electrons.
It can be seen from symmetry arguments that all the remaining
perturbation integrals vanish; we shall discuss /Jls(l) 2s(2)
e 2
— ls(l) 2p*(2) dridri as an example. In this integral the func
^12
tion 2p x (2) is an odd function of the coordinate x 2 , and inasmuch
as all the other terms in the integrand are even functions of a* 2 ,
the integral will vanish, the contribution from a region with x 2
negative canceling that from the corresponding region with x 2
positive.
The solution of Equation 292 leads to the perturbation energy
values
W = J. + K.,
J v + K Pf (triple root), ( (2 *^ )
J p — K p , (triple root).j
The splitting of the unperturbed level represented by these
equations is shown in Figure 291.
Vin29a]
THE HELIUM ATOM
213
One part of the splitting, due to the difference of the Coulomb
integrals J 9 and J p , can be easily interpreted as resulting from the
difference in the interaction of an inner Is electron with an
outer 2s electron or 2p electron. This effect was recognized in
the days of the old quantum theory, being described as resulting
from greater penetration of the core of the atom (the nucleus
plus the inner electrons) by the more eccentric orbits of the
W
ni,ryl,e
Fig. 291. — The splitting of energy levels for the helium atom.
outer electron, with a consequent increase in stability, an s orbit
being more stable than a p orbit with the same value of n, and
so on. 1 (It is this dependence of the energy of an electron on I
as well as n which causes the energy levels of an atom to depend
largely on the electronic configuration, this expression meaning
the n and I values of all electrons. These values are usually
indicated by writing ns, np 7 etc., with the number of similar
electrons indicated by a superscript. Thus Is 2 indicates two Is
electrons, ls 2 2p these plus a 2p electron, and so on.)
1 Pauling and Goudsmit, "The Structure of Line Spectra," Chap. Ill
214 THE SPINNING ELECTRON [VIH29b
On the other hand, the further splitting due to the integrals
K, and K p was not satisfactorily interpreted before the develop
ment of the quantum mechanics. It will be shown in Section 41
that we may describe it as resulting from the resonance phe
nomenon of the quantum mechanics. The zerothorder wave
function for the state with W = J 8 + K s , for example, is
i {1«(1)2*(2)+ 2 5 (1)1 5 (2)1;
the atom in this state may be described as resonating between the
structure in which the first electron is in the Is orbit and the
second in the 2s orbit and that in which the electrons have been
interchanged.
A wave function of the type just mentioned is said to be
symmetric in the positional coordinates of the two electrons, inasmuch
as the interchange of the coordinates of the two electrons leaves
the function unchanged. On the other hand, the wave function
~^{ls(l)2s(2) 2«(1) ls(2)}
is antisymmetric in the positional coordinates of the electrons,
their interchange causing the function to change sign. It is
found that all wave functions for a system containing two
identical particles are either symmetric or antisymmetric in the
coordinates of the particles.
For reasons discussed in the next section, the stationary states
of twoelectron atoms represented by symmetric and by anti
symmetric positional wave functions are called singlet states
and triplet states, respectively. The triplet state from a given
configuration is in general more stable than the singlet state.
29b. The Consideration of Electron Spin. The Pauli Exclu
sion Principle. — In reconsidering' the above perturbation prob
lem, taking cognizance of the spin of the electrons, we must
deal with thirtytwo initial spinorbit wave functions instead of
the eight orbital functions ls(l) 2s(2), ls(l) 2p*(2), etc. These
thirtytwo functions are obtained by multiplying each of the
eight orbital functions by each one of the four spin functions
a(l) a(2),
<*(D 0(2),
0(1) «(2),
0(1) 0(2).
Vin29b]
THE HELIUM ATOM
215
Instead of using the second and third of these, it is convenient to
use certain linear combinations of them, taking as the four spin
functions for two electrons
«(1) «(2),
^={a(l)0(2)+/9(l)a(2)} f j
0(1) 0(2),
i«(l)l8(2) 0(1)«(2)
(2&5)
1
V2
These are normalized and mutually orthogonal. The first
three of them are symmetric in the spin coordinates of the two
electrons, and the fourth is antisymmetric. It can be shown that
these are correct zerothorder spin functions for a perturbation
involving the spins of the two electrons.
Taking the thirtytwo orbit functions in the order
ls(l) 28(2) a(l) a(2),
28(1) 18(2) o(l) o(2),
18(1) 2p x (2) «(1) a(2),
1«(1) 28(2) • 7^{«(D /?(2) + /?(!) a(2)},
obtained by multiplying the eight orbital functions by the first
spin function, then by the second spin function, and so on,
we find that the secular equation has the form
= 0,
216 THE SPINNING ELECTRON [Vni29b
in which each of the small squares is an eightrowed determinant
identical with that of Equation 292. The integrals outside of
these squares vanish because of the orthogonality of the spin
functions and the nonoccurrence of the spin coordinates in the
perturbation function e 2 /r X2 . The roots of this equation are
the same as those of Equation 292, each occurring four times,
however, because of the four spin functions.
The correct zerothorder wave functions are obtained by
multiplying the correct positional wave functions obtained in the
preceding section by the four spin functions. For the con
figuration ls2s alone they are
L{1*(1) 2s(2) + 2 5 (1) 1«(2)} • «(1) a(2),
7^{1«(1) 2s(2) + 2«(1) 1«(2)} • 7^{«(1) /8(2) + 0(1) «(2)}
;^§{1«(1) 2»(2) + 2^(1) 1«(2)} • 0(1) 0(2),
4={1«(1) 25(2)  25(1) 15(2)} • 4={«(1) 0(2) ~ 0(1) «(2)},
f~L{l5(l) 25(2)  25(1) 15(2)} . «(1) a(2),
Triplet^ 1 *^ 2S(2)  2S(1) 1S(2)} ' ^ { ^ (1) ^ (2) +
1 {15(1)25(2) 2s(l)l5(2)}0(l)0(2),
0(1) «(2)}
^V2
Singlet 7^{1«(1) &(2) + 25(1) ls(2)} • 4={«(1) 0(2) 
0(l)a(2).
Of these eight functions, the first four are symmetric in the
coordinates of the two electrons, the functions being unchanged
on interchanging these coordinates. This symmetric character
results for the first three functions from the symmetric character
of the orbital part and of the spin part of each function. For
the fourth function it results from the antisymmetric character
of the two parts of the function, each of which changes sign on
interchanging the two electrons.
Vin29b] THE HELIUM ATOM 217
The remaining four functions are antisymmetric in the two
electrons, either the orbital part being antisymmetric and the
spin part symmetric, or the orbital part symmetric and the spin
part antisymmetric.
Just as for \s2s, so each configuration leads to some symmetric
and some antisymmetric wave functions. For Is 2 , for example,
there are three of the former and one of the latter, obtained by
combining the symmetric orbital wave function of Section 236
with the four spin functions. For ls2p there are twelve of each
ls?p
Symmetric Antisymmetric
• • •
• • •
• • •
• •• 01s2p'P
• • •
O O O • • • ls2p 3 P
• ••
O ls2» l S
• • #ls2s*S
Is 2 • • • Ols 2 'S
Fig. 292. — Levels for configurations la 1 , ls2a, and la2p of the helium atom. $,
spinsymmetric wave functions; Ot spinantisymmetric wave functions
type, nine spinsymmetric and orbitalsymmetric, three spin
antisymmetric and orbitalantisymmetric; nine spinsvmmetric
and orbitalantisymmetric, and three spinantisymmetric and
orbitalsymmetric. The levels thus obtained for the helium
atom by solution of the wave equation are shown in Figure 292,
the completely symmetric wave functions being represented
on the left and the completely antisymmetric ones on the right.
Now it can be shown that if a helium atom is initially in a
symmetric state no perturbation whatever will suffice to cause
it to change to any except symmetric states (the two electrons
being considered to be identical). Similarly, if it is initially
in an antisymmetric state it will remain in an antisymmetric
state. The solution of the wave equation has provided us with
218 THE SPINNING ELECTRON [VIII29b
two completely independent sets of wave functions. To show
that no perturbation will cause the system in a state represented
by the symmetric wave function yp s to change to a state repre
sented by the antisymmetric wave function ^a we need only
show that the integral
vanishes (R' being the perturbation function, involving the spin
as well as the positional coordinates of the electrons), inasmuch
as it is shown in Chapter XI that the probability of transition
from one stationary state to another as a result of a perturbation
is determined by this integral. Now, if the electrons are identi
cal, the expression H'fa is a symmetrical function of the coordi
nates, whereas \p% is antisymmetric; hence the integrand will
change sign on interchanging the coordinates of the two electrons,
and since the region of integration is symmetrical in these
coordinates, the contribution of one element of configuration
space is just balanced by that of the element corresponding
to the interchange of the electrons, and the integral vanishes. 1
The question as to which types of wave functions actually
occur in nature can at present be answered only by recourse to
experiment. So far all observations which have been made on
helium atoms have shown them to be in antisymmetric states. 2
We accordingly make the additional postulate that the wave
function representing an actual state of a system containing two
or more electrons must be completely antisymmetric in the coordinates
of the electrons; that is, on interchanging the coordinates of any
two electrons it must change its sign. This is the statement of
the Pauli exclusion principle in wavemechanical language.
This is a principle of the greatest importance. A universe
based on some other principle, that is, represented by wave
functions of different symmetry character, would be completely
different in nature from our own universe. The chemical
properties in particular of substances are determined by this
principle, which, for example, restricts the population of the
K shell of an atom to two electrons, and thus makes lithium
1 The same conclusion is reached from the following argument: On inter
changing the subscripts 1 and 2 the entire integral is converted into itself
with the negative sign, and hence its value must be zero.
2 The states are identified through the splitting due to spinorbit inter
actions neglected in our treatment.
Vin29b] THE HELIUM ATOM 219
an alkali metal, the third electron being forced into an outer
shell where it is only loosely bound.
To show this, we may mention that if A represents a spinorbit
function for one electron (such that A(l) = ls(l) a(l), for
example) and B, C, • • • , E others, then the determinantal
function
A{\) B(l) . . . E(\)
A{2) 5(2) . . . E(2)
A(N) B(N) . . . E(N)
is completely antisymmetric in the N electrons, and hence a
wave function of this form for the iVelectron system satisfies
Paulas principle, since from the properties of determinants the
interchange of two rows changes the sign of the determinant.
Moreover, no two of the functions A, B } • • • , E can be equal,
as then the determinant would vanish. Since the only spinorbit
functions based on a given oneelectron orbital function are the
two obtained by multiplying by the two spin functions a and 0,
we see that no more than two electrons can occupy the same orbital
in an atom, and these two must have their spins opposed; in other
words, no two electrons in an atom can have the same values of
the four quantum numbers w, I, ra, and m 8 . Paulas original
statement 1 of his exclusion principle was in nearly this language;
its name is due to its limitation of the number of electrons in an
orbit.
The equations of quantum statistical mechanics for a system of
nonidentical particles, for which all solutions of the wave
equations are accepted, are closely analogous to the equations
of classical statistical mechanics (Boltzmann statistics). The
quantum statistics resulting from the acceptance of only anti
symmetric wave functions is considerably different. This
statistics, called FermiDirac statistics, applies to many problems,
such as the PauliSommerfeld treatment of metallic electrons
and the ThomasFermi treatment of manyelectron atoms.
The statistics corresponding to the acceptance of only the
completely symmetric wave functions is called the BoseEinstein
statistics. These statistics will be briefly discussed in Section 49.
1 W. Pauli, Z. f. Phys. 31, 765 (1925).
220 THE SPINNING ELECTRON [Vm29b
It has been found that for protons as well as electrons the wave
functions representing states occurring in nature are antisym
metric in the coordinates of the particles, whereas for deuterons
they are symmetric (Sec. 43/).
The stationary states of the helium atom, represented on the
right side of Figure 292, are conveniently divided into two sets,
shown by open and closed circles, respectively. The wave
functions for the former, called singlet states, are obtained by
multiplying the symmetric orbital wave functions by the single
antisymmetric spin function — j=.{a(l) 0(2) — 0(1) «(2)}.
V2
Those for the latter, called triplet states, are obtained by mul
tiplying the antisymmetric orbital wave functions by the three
symmetric spin functions. 1 The spinorbit interactions which
we have neglected cause some of the triplet levels to be split
into three adjacent levels. Transitions from a triplet to a
singlet level can result only from a perturbation involving the
electron spins, and since interaction of electron spins is small
for light atoms, these transitions are infrequent; no spectral line
resulting from such a transition has been observed for helium.
It is customary to represent the spectral state of an atom by
a term symbol such as l S, Z S, Z P, etc. Here the superscript on
the left gives the multiplicity, 1 signifying singlet and 3 triplet.
The letters S, P, etc., represent the resultant of the orbital
angularmomentum vectors of all the electrons in the atom.
This is also given by a resultant azimuthal quantum number L,
the symbols S,P,D,F, • • • corresponding to L = 0,1,2,3, • * • .
If all the electrons but one occupy s orbitals, the value of L is
the same as that of I for the odd electron, so that for helium the
configurations Is 2 , ls2s, and ls2p Jead to the states l S, l S and 3 >S,
and l P and 3 P. Use is also made of a resultant spin quantum
number S (not to be confused with the symbol S for L = 0),
1 The electrons are often said to have their spins opposed or antiparallel
in singlet states and parallel in triplet states, the spin function
^{« (1)0(2) +0(1) «(2)}
in the latter case representing orientation of the resultant spin with zero
component along the z axis.
Vm29c] THE HELIUM ATOM 221
which has the value for singlet states and 1 for triplet states,
the multiplicity being equal to 2S + l. 1
The results which we have obtained regarding the stationary
states of twoelectron atoms may be summarized in the following
way. The main factors determining the term values are the
values of the principal quantum numbers fti and n 2 for the
two electrons and of the azimuthal quantum numbers h and h t
smaller values of these numbers leading to greater stability.
These numbers determine the configuration of the atom. The
configuration Is 2 leads to the normal state, ls2s to the next most
stable states, then ls2p, and so on. For configurations with
n\l\ different from n 2 ?2 there is a further splitting of the levels
for a given configuration, due to the resonance integrals,
leading to singlet and triplet levels, and to levels with different
values of the resultant azimuthal quantum number L in case
that both h and Z 2 are greater than zero. The triplet levels may
be further split into their finestructure components by the
spinorbit interaction, which we have neglected in our treatment.
It is interesting to notice that these interactions completely
remove the degeneracy for some states, such as ls2s l S,
but not for others, such as ls2s Z S, which then show a further
splitting (Zeeman effect) on the application of a magnetic field
to the atom.
Problem 291. Evaluate the integrals J and K for ls2s and ls2p of
helium, and calculate by the firstorder perturbation theory the term values
for the levels obtained from these configurations. Observed term values
(relative to He + ) are ls2s l S 32033, U2s *S 38455, ls2p l P 27176, and l«2p 3 P
29233 cm" 1 .
29c. The Accurate Treatment of the Normal Helium Atom. —
The theoretical calculation of the energy of the normal helium
atom proved to be an effective stumbling block for the old
quantum theory. On the other hand, we have already seen that
even the first attack on the problem by wavemechanical methods,
1 For a detailed discussion of spectroscopic nomenclature and the vector
model of the atom see Pauling and Goudsmit. "The Structure of Line
Spectra." The triplet levels of helium were long called doublets, complete
resolution being difficult. Their triplet character was first suggested by
J. C. Slater, Proc. Nat. Acad. Set. 11, 732 (1925), and was soon verified
experimentally by W. V. Houston, Phys. Rev. 29, 749 (1927). The names
parhelium and orthohelium were ascribed to the singlet and triplet levels,
respectively, before their nature was understood.
222 THE SPINNING ELECTRON [Vm29c
the firstorder perturbation treatment given in Section 236,
led to a promising result, the discrepancy of about 4 v.e. (accept
ing the experimental value as correct) being small compared
with the discrepancies shown by the oldquantumtheory
calculations. It is of interest to see whether or not more and
more refined wavemechanical treatments continue to diminish
the discrepancy with experiment and ultimately to provide a
theoretical value of the ionization potential agreeing exactly
with the experimental (spectroscopic) value 24.463 v. 1 The
success of this program would strengthen our confidence in our
wavemechanical equations, and permit us to proceed to the
discussion of manyelectron atoms and molecules.
No exact solution of the wave equation has been made, and all
investigators have used the variation method. 2 The simplest
extension of the zerothorder wave function e~ 2 *, with s = (ri + r 2 )/
a , is to introduce an effective nuclear charge Z'e in place of the
true nuclear charge 2e in the wave function. This function, e~ z ' 9 ,
minimizes the energy when the atomic number Z' has the value
2 K6> corresponding to a screening constant of value %$ (Sec.
266). The discrepancy with the observed energy 3 (Table 291)
is reduced by this simple change to 1.5 v.e., which is onethird the
discrepancy for UnsolcTs treatment. This wave function cor
responds to assuming that each electron screens the other
1 Calculated from Lyman's term value 198298 cm"" 1 corrected by Paschen
to 198307.9 cm"" 1 ; T. Lyman, Astrophys. J. 60, 1 (1924); F. Paschen, Sitzber.
preuss. Akad. Wiss. 1929, p. 662.
2 The principal papers dealing with the normal helium atom are A. Unsold,
Ann. d. Phys. 82, 355 (1927); G. W. Kellner, Z.f. Phys. 44, 91, 110 (1927);
J. C. Slater, Proc. Nat. Acad. Set. 13, 423 (1927); Phys. Rev. 32, 349 (1928);
C. Eckart, Phys. Rev. 36, 878 (1930); E. A. Hylleraas, Z. f. Phys. 48, 469
(1928); 54, 347 (1929); 65, 209 (1930). A summary of his work is given by
Hylleraas in Skrifter det Nor she Vid.Ak. Oslo, I. Mat.Naturv. Klasse 1932,
pp. 5141. For the special methods of evaluating and minimizing the
energy integral, the reader is referred to these papers.
3 The experimental value —78.605 v.e. = —5.8074 R^hc for the energy
of the normal helium atom is obtained by adding to the observed first
ionization energy 24.463 v,e. (with the minus sign) the energy
4R Kt hc  54.1416 v.e.
of the helium ion. Hylleraas has shown that the correction for motion of
the nucleus in the neutral helium atom is to be made approximately by
using i^HeJ that is, by assigning to each electron the reduced mass with the
helium nucleus.
Vin29c] THE HELIUM ATOM 223
from the nucleus in the same way as a charge —%ee on the
nucleus.
Problem 292. (a) Calculate approximately the energy the normal
lithium atom would have if the allowed wave functions were completely
symmetric in the electrons, using for the positional wave function the
product function ls(l) ls(2) ls(3), in which Is contains the effective nuclear
charge Z' = 3 — S, and minimizing the energy relative to Z' or S. From
this and a similar treatment of Li+ obtain the first ionization energy. The
observed value is 5.368 v.e. (b) Obtain a general formula for the Nth
ionization energy of an atom with atomic number Z in such a BoseEinstein
universe, using screeningconstant wave functions. Note the absence of
periodicity in the dependence on Z.
We might now consider other functions of the type F(ri)F(r 2 ) y
introducing other parameters. This has been done in a general
way by Hartree, in applying his theory of the selfconsistent
field (Chap. IX), the function F(r x ) being evaluated by special
numerical and graphical methods. The resulting energy value,
as given in Table 291, is still 0.81 v.e. from the experimental
value. Even the simple algebraic function
Z\r\ Zyr\ Ztri Zyrt
leads to as good a value of the energy. (This is function 4 of
the table, there expressed in terms of the hyperbolic cosine.)
This variation function we may interpret as representing one
electron in an inner orbit and the other in an outer orbit, the
values of the constants, Z x = 2.15 and Z 2 = 1.19, corresponding
to no shielding (or, rather, a small negative shielding) for the
inner electron by the outer, and nearly complete shielding for
the outer electron by the inner. By taking the sum of two
terms the orbital wave function is made symmetric in the two
electrons. It is interesting that the still simpler function
5 leads to a slightly better value for the energy. Various more
complicated functions of s and t were also tried by Kellner and
Hylleraas, with considerable improvement of the energy value.
Then a major advance was made by Hylleraas by introducing
in the wave function the coordinate u = rn/a 0t which occurs
in the interaction term for the two electrons. The simple two
parameter functions 6 and 7 provide values of the energy of the
atom accurate to H P er cen ^. Here again the polynomial in u
is more satisfactory than the more complicated exponential
224
THE SPINNING ELECTRON
[VIII29C
function, suggesting that a polynomial factor containing further
powers of u } t, and s be used. The functions 8, 9, and 10 show
that this procedure leads quickly to a value which is only slightly
changed by further terms, the last three terms of 10 being
reported by Hylleraas as making negligible contributions.
The final theoretical value for the energy of the helium atom is
0.0016 v.e. below the experimental value. Inasmuch as this
theoretical value, obtained by the variation method, should
be an upper limit, the discrepancy is to be attributed to a numer
ical error in the calculations or to experimental error in the
ionization energy, or possibly to some small effects such as
electronspin interactions, motion of the nucleus, etc. At
any rate the agreement to within 0.0016 v.e. may be considered
as a triumph for wave mechanics when applied to manyelectron
atoms.
Table 291. — Variation Functions for the Normal Helium Atom 1
Symbols: s = » t = —
do
Experimental value of W = 5.80736/2 H «^c
— r 2 r 12
1 u = —
ao cio
Variation function, with best values of
constants 2
1. 6" 2 *
2. e~ z '% Z' = 27 / 16 = 1.6875
3. F(r x )F(r 2 )
4. «*'• cosh ct, Z' = t.67, c = 0,48. . .
5. e~ z, '(l + c 2 < 2 ), Z' = 1.69, c 2 = 0.142. . .
6. e*V u , Z' = 1.86, c = 0.26
7. e~ z '»(l + citi), Z' = 1.849, c, = 0.364. .
8. e~ zf '(l h cxu 4 c 2 t 2 )
Z' = 1.816, c, = 0.30, c 2 = 0.13.
9. e~ z, '{\ + ciu + c 2 t 2 + c 3 s + c 4 s 2 f c 6 w 2 )
Z' = 1.818, ci = 0.353, c 2 = 0.128,
c,  0.101, ca = 0.033, c 6 = 0.032
10. e~ z '* (polynomial with fourteen terms) . .
Energy,
in units
— RnJic
50
6953
75
7508
7536
7792
7824
80488
5.80648
5.80748
Difference with
experiment
Units
—Rjuhc
0.31
0.1120
0.06
0.0565
0.0537
0.0281
0.0249
0.00245
0.00085
0.00012
V.e.
4.19
1.53
0.81
0.764
0.726
0.380
0.337
0.0332
0115
0.0016
1 A few variation functions which have been tried are not included in the table because
they are only slightly better than simpler ones; for example, the function e~*'* (1 — ae~ c n u ),
which is scarcely better than function 6. (D. R. Hartree and A. L. Ingman, Mem.
Manchester Phil. Soc. 77, 69 (1932).)
5 The normalization factor is omitted. Of these functions, 1 is due to Unsold, 2 to Kell
ner, 3 to Hartree and Gaunt, 4 to Eckart and Hylleraas, and the remainder to Hylleraas.
VIII29d]
THE HELIUM ATOM
225
Hylleraas's masterly attack on the problem of the energy of
normal helium and heliumlike ions culminated in his derivation
of a general formula for the first ionization energy J of these
atoms and ions. 1 This formula, obtained by purely theoretical
considerations, is
/ =
RJic
1 +
— (z>  \l
m Q y 4
M
Z + 0.31488
0.01752 0.00548 \
z + z* )'
(296)
in which M is the mass and Z the atomic number of the atom.
Values calculated by this formula 2 are given in Table 292,
together with experimental values obtained spectroscopically,
mainly by Edl£n 3 and coworkers. It is seen that there is agree
Table 292. — Ionization Energies of Twoelectron Atoms
Atom
/ calculated, v.e.
/ observed, v.e.
H
0.7149
He
24.465
24.463
Li+
75.257
75.279 ± 0.012
Be + +
153.109
153.09 ±0.10
B+++
258.029
258.1 ±0.2
C 4 +
390.020
389.9 ±0.4
N 5 +
549.085
Q6+
735.222
ment to within the experimental error. Indeed, the calculated
values are now accepted as reliable by spectroscopists. 4
Included in the table is the value 0.7149 v.e. for the ionization
energy of the negative hydrogen ion H~~. This shows that the
hydrogen atom has a positive electron affinity, ^amounting to
16480 cal/mole. The consideration of the crystal energy of the
alkali hydrides has provided a rough verification of this value.
29d. Excited States of the Helium Atom. — The variation
method can be applied to the lowest triplet state of helium as
well as to the lowest singlet state, inasmuch as (neglecting
iE. A. Hylleraas, Z.f. Phys. 66, 209 (1930).
2 Using 1 v.e. = 8106.31 cm" 1 and R* = 109737.42 cm" 1 .
3 A. Ericson and B. Edl6n, Nature 124, 688 (1929); Z. f. Phys. 69, 666
(1930); B. Edl£n, Nature 127, 405 (1930).
* B. Edl£n, Z. f. Phys. 84, 746 (1933).
226 THE SPINNING ELECTRON [VIII29e
spinorbit interactions) the triplet wave functions are anti
symmetric in the positional coordinates of the two electrons,
and contain no contribution from singlet functions (Sec. 26c).
A simple and reasonable variation function is
1«*'(1) 2s*"(2)  2sz(l) lsz<(2),
in which ls Z ' and 2s Z " signify hydrogenlike wave functions with
the indicated effective nuclear charges as parameters. We
would expect the energy to be minimized for Z r = 2 and Z" = 1.
Calculations for this function have not been made. However,
Hylleraas 1 has discussed the function
se z ' a sinh ct, (297)
obtaining the energy value — 4.342(XR He Ac, not far above the
observed value — 4.3504 J R He /ic. This function is similar to the
hydrogenlike function (containing some additional terms),
and the parameter values found, Z' = 1.374 and c = 0.825,
correspond to the reasonable values Z' = 2.198 and Z" = 1.099.
Hylleraas has also replaced s in 297 by s + CiU, obtaining the
energy — 4.3448# H ^c, and by s + c 2 £ 2 , obtaining the energy
— 4.3484#H e Ac. It is probable that the series s + CiU + c 2 t 2
would lead to very close agreement with experiment.
Numerous investigations by Hylleraas and others 2 have
shown that wave mechanics can be applied in the treatment of
other states of the helium atom. We shall not discuss further
the rather complicated calculations.
29e. The Polarizability of the Normal Helium Atom.— A
quantity of importance for many physical and chemical con
siderations (indices of refraction, electric dipole moments,
term values of nonpenetrating orbits, van der Waals forces,
etc.) is the polarizability of atoms and molecules, mentioned in
Problem 261 and Sections 27a and 27e. We may write as the
energy of a system in an electric field of strength F the expression
W = W°  %aF* + • • • (298)
1 E. A. Hylleraas, Z. j. Phys. 64, 347 (1929).
2 W. Heisenberg, Z. f. Phys. 39, 499 (1926); A. Unsold, Ann. d. Phys.
82, 355 (1927); E. A. Hylleraas and B. Undheim, Z.f. Phys. 66, 759 (1930);
E. A. Hylleraas, ibid. 66, 453 (1930); 83, 739 (1933); J. P. Smith, Phys.
Rev. 42, 176 (1932); etc.
VIII29e] THE HELIUM ATOM 227
in case that the term linear in F vanishes, the permanent electric
moment of the system being zero. The electric moment induced
in the system by the field is aF } the factor of proportionality a.
being called the polarizability. The polarizability of the
molecules in a gas determines its index of refraction n (for light
of very large wave length) and its dielectric constant D, according
to the equation
N 3 n 2 l 3 P 1 , 9Q _ Q .
V a =^tf + 2 = ±*D+2' (29 ~ 9)
in which N is Avogadro's number and V is the molal volume of
the substance. The mole refraction R is defined as
R = —a = 2.54 • 10 24 <*. (2910)
The dimensions of R and a are those of volume, and their magni
tudes are roughly those of molal volumes and molecular volumes,
respectively; for example, for monatomic hydrogen 22 = 1.69 cm 3
and a = 0.667 • 10~ 24 cm 3 (Sec. 27a). Values of R and a are
determined experimentally mainly by measurement of indices
of refraction and of dielectric constants, 1 rough values being
also obtainable from spectral data. 2
The value of the polarizability a of an atom or molecule can
be calculated by evaluating the secondorder Stark effect energy
— %aF 2 by the methods of perturbation theory or by other
approximate methods. A discussion of the hydrogen atom has
been given in Sections 27a and 27 e (and Problem 261). The
helium atom has been treated by various investigators by the
variation method, and an extensive approximate treatment
of manyelectron atoms and ions based on the use of screening
constants (Sec. 33a) has also been given. 3 We shall discuss the
variational treatments of the helium atom in detail.
The additional term in the Hamiltonian due to the electric
1 The total polarization of a gas may be due to polarization of the electrons
in the gas molecules (for fixed nuclear positions), polarization of the nuclei
(with change in the relative positions of the nuclei in the molecules), and
orientation of molecules with permanent electric dipole moments. We are
here discussing only the first of these mechanisms; the second is usually
unimportant, and the third is treated briefly in Section 49/.
1 See Pauling and Gotjdsmit, "The Structure of Line Spectra," Sec. 11.
8 L. Paulino, Proc. Roy. Soc. A114, 181 (1927).
228
THE SPINNING ELECTRON
[Vin29e
field (assumed to lie along the z axis) is eF{z x + z 2 ), z x and z 2
being the z coordinates of the two electrons relative to the nucleus.
The argument of Section 27c suggests that the variation function
be of the form
* = *°{1 + («i + *0/(*i, Vi, *u **, 2/2, 22)}, (2911)
in which xp is an approximate wave function for zero field.
Variation functions of this form (or approximating it) have been
discussed by Hass6, Atanasoff, and Slater and Kirkwood, 1
whose results are given in Table 293.
Table 293. — Variation Functions for the Calculation of the
pol ariz ability of the normal helium atom
Experimental value: a = 0.205 • 10"" 24 cm 3
r\ + r 2
s = ■
Variation function
a
References 1
1.
e*''l f A(zi + z 2 )\
0.15010~ 24 cm 3
H
2.
e z>*\i __ A(zie~ z " r i + z 2 e z " r z)}
.164
SK
3.
(rirj)° 1M e^«{l + A(zie" z " r i + zi<r z " r ')\
.222
SK
4.
e z ''[l +A(zi+ z 2 ) + B{z l r l + z 2 r 2 )) .. . .
.182
H
5.
6 z't(i 1^4(2! f 2 2 ) f terms to quartic)}
.183
H
6.
e z '*(l +c lU )[l +A( 2i +z 2 ) +
Bfari f z 2 r 2 ))
.201
H
7.
e~ z '>\\ +au + c 2 t 2 +A(zi +22)}
.127
A
8.
e~ z '*\l + citt + c 2 * 2 + W + £a)(zi + z*)
+ Cf(2l ~ *«)}
.182
A
9.
gZ'« j 1 + citt + c 2 * 2 + c 8 s + c 4 s 2 + c 5 w 2
f (A f B«)(«i + 22) + Ct( Zl  z 2 )
+ DU( ZI +Z2)
.194
A
10.
<r*'*(l + Cl w + c 2 * 2 )(l +A(Zi+ z 2 )
+B(*,r, + * 2 r 2 )}
.231
H
11.
A nonalgebraic function
.210
SK
1 H — Hass6, A  Atanasoff, SK = Slater and Kirkwood.
Of these functions, 1, 2, 4, and 5 are based on the simple
screeningconstant function 2 of Table 291; these give low
values of a, the experimental value (from indices of refraction
extrapolated to large wave length of light and from dielectric
1 H. R. Hasse, Proc. Cambridge Phil. Soc. 26, 542 (1930), 27, 66 (1931);
J. V. Atanasoff, Phys. Rev. 36> 1232 (1930); J. C. Slater and J. G. Kirk
wood, Phys. Rev. 37, 682 (1931).
Vin29e1 THE HELIUM ATOM 229
constant) being about 0.205 • 10~ 24 cm 3 . The third function,
supposed to provide a better approximation to the correct wave
function for large values of r\ and r 2 (that is, in the region of the
atom in which most of the polarization presumably occurs),
overshoots the mark somewhat. (The fundamental theorem
of the variation method (Sec. 26a) does not require that a
calculation such as these give a lower limit for a, inasmuch as
the wave function and energy value for the unperturbed system
as well as for the perturbed system are only approximate.)
Function 6 is based on 7 of Table 291, 7, 8, and 10 on 8, and
9 on 9. It is seen that functions of the form 2911 (6, 10) seem
to be somewhat superior to functions of the same complexity
not of this form (7, 8, 9). Function 11 is based on a helium
atom function (not given by a single algebraic expression) due to
Slater. 1
It is seen that the values of a given by these calculations in
the main lie within about 10 per cent of the experimental value 2
0.205 • 10~ 24 cm 3 . For Li+, Hass6, using function 6, found the
value a = 0.0313 • 10~ 24 cm 3 ; the only other values with which
this can be compared are the spectroscopic value 3 0.025 and the
screeningconstant value 2 0.0291 • 10~ 24 cm 3 .
Problem 293. Using the method of Section 27e and the screening
constant wave function 2 of Table 291, evaluate the polarizability of the
helium atom, taking as the zero point for energy the singly ionized atom.
1 J. C. Slater, Phys. Rev. 32, 349 (1928).
2 The rough screeningconstant treatment mentioned above gives the
values 0.199 • 10" 24 cm 3 for He and 0.0291 • 10" 24 cm 3 for Li+.
3 J. E. Mayer and M. G. Mayer, PJiys. Rev. 43, 605 (1933).
CHAPTER IX
MANYELECTRON ATOMS
Up to the present time no method has been applied to atoms
with more than two electrons which makes possible the computa
tion of wave functions or energy levels as accurate as those for
helium discussed in Section 29c. With the increasing complexity
of the atom, the labor of making calculations similar to those
used for the ground state of helium increases tremendously.
Nevertheless, many calculations of an approximate nature have
been carried out for larger atoms with results which have been
of considerable value. We shall discuss some of these in this
chapter. 1
30. SLATER'S TREATMENT OF COMPLEX ATOMS
30a. Exchange Degeneracy. — All of the methods which we
shall consider are based on a first approximation in which the
interaction of the electrons with each other has either been
omitted or been replaced by a centrally symmetric field approxi
mately representing the average effect of all the other electrons
on the one under consideration. We may first think of the prob
lem as a perturbation problem. The wave equation for an atom
with N electrons and a stationary nucleus is
*l \ t' = l t,j>i I
in which r, is the distance of the ith electron from the nucleus,
Ta is the distance between the ith and jth electrons, and Z is
the atomic number.
If the terms in r# are omitted, this equation is separable into
N threedimensional equations, one for each electron, just as
was found to be the case for helium in Section 236. To this
1 This chapter can be omitted by readers not interested in atomic spectra
and related subjects; however, the treatment is closely related to that for
molecules given in Chapter XIII.
230
IX30a] SLATER'S TREATMENT OF COMPLEX ATOMS 231
degree of approximation the wave function for the atom may be
built up out of singleelectron wave functions; that is, a solution
of the equation for the atom with 2e 2 /r# omitted is
*! = Ua(l) u*(2) • • • u,(N), (302)
in which t/«(l), etc., are the solutions of the separated single
electron equations with the three quantum numbers 1 symbolized
by a, ($, • • • , v and the three coordinates symbolized by
1, 2, • • • , N. With this form for ^° the individual electrons
retain their identity and their own quantum numbers. How
ever, an equally good solution of the unperturbed equation cor
responding to the same energy as Equation 302 is
+° = ua(2) UfiO) • • • u v (N) } (303)
in which electrons 1 and 2 have been interchanged. In general,
the function
l# = Pu a (l) Ufi(2) • • • u v (N), (304)
in which P is any permutation of the electron coordinates, is an
unperturbed solution for this energy level.
The meaning of the operator P may be illustrated by a simple example.
Let us consider the permutations of the three symbols x i} z 2 , x%. These are
Xi t Xi, Xz) x 2 , x%> x\) Xz, Xi, x 2 ; x 2 , xi, x z ; Xi, x z , x%\ x z , x 2 , x\. Any one of these
six may be represented by Px i, x 2 , x Z) in which P represents the operation of
permuting the symbols x h x 2 , x z in one of the above ways. The operation P
which yields xi, x 2 , x z is called the identity operation.
Any of the above permutations can be formed from x h x 2 , x z by successive
interchanges of pairs of symbols. This can be done in more than one way,
but the number of interchanges necessary is either always even or always
odd, regardless of the manner in which it is carried out. A permutation
is said to be even if it is equivalent to an even number of interchanges, and
odd if it is equivalent to an odd number. We shall find it convenient to
use the symbol ( — l) p to represent +1 when P is an even permutation and
— 1 when P is an odd permutation.
Multiplication of the operators P and P' means that P and P' are to be
applied successively. The set of all the permutations of N symbols has the
property that the product PP' of any two of them is equal to some other
permutation of the set. A set of operators with this property is called 8
group, if in addition the set possesses an identity operation and if every
operation P possesses an inverse operation P" 1 such that PP" 1 is equivalent
to the identity operation. There are #! permutations of N different
symbols.
1 The symbols a, ft • • • , v are of course not related to the spin functions
a and 0.
232 MANYELECTRON ATOMS [IX30a
At this point we may introduce the spin of the electrons into
the wave function (in the same manner as for helium) by multi
plying each singleelectron orbital function by either a(o>) or
0(co). For convenience we shall include these spin factors in
the functions u a (l), etc., so that hereafter a, j8, 7, • • • represent
four quantum numbers n, I, mi, and m 8 for each electron and 1,
2, • • • represent four coordinates r», & iy <p», and co». As discussed
in Section 29a for the twoelectron case, treatment of this
degenerate energy level by perturbation theory (the electron
interactions being the perturbation) leads to certain combinations
r = 77Fi^ cpPUaW ^ (2) ' ' ' Uv(N) (3a ~ 5)
p
for the correct zerothorder normalized wave functions. One
of these combinations will have the value +1 for each of the
coefficients c P . Interchange of any pair of electrons in this
function leaves the function unchanged; i.e., it is completely
symmetric in the electron coordinates. For another combina
tion the coefficients cp are equal to +1 or to — 1, according as P
is an even or an odd permutation. This combination is com
pletely antisymmetric in the electrons; i.e., the interchange of
any two electrons changes the sign of the function without
otherwise altering it. Besides these two combinations, which
were the only ones which occurred in helium, there are for
manyelectron atoms others which have intermediate symmetries.
However, this complexity is entirely eliminated by the appli
cation of the Pauli exclusion principle (Sec. 29b) which says that
only the completely antisymmetric combination
r = ^2 (_1)i>Pwa(1) M " (2) ' ■ ■ u  {N) (30_6)
p
has physical significance. This solution may also be written as a
determinant,
U a (l) Ufi(l) . . . U v {\)
u a {2) u (2) . . . tt„(2)
^0
VW~\
(307)
Ua(N) u,(N) . . . u v (N)
as was done in Section 296. The two forms are identical.
IX30b] SLATER'S TREATMENT OF COMPLEX ATOMS 233
30b. Spatial Degeneracy. — In the previous section we have
taken care of the degeneracy due to the N ! possible distributions;
of the N electrons in a fixed set of N functions u. There still
remains another type of degeneracy, due to the possibility of
there being more than one set of spinorbit functions correspond
ing to the same unperturbed energy. In particular there may
be other sets of u's differing from the first in that one or more of
the quantum numbers mi or m 8 have been changed. These
quantum numbers, which represent the z components of orbital
and spin angular momentum of the individual electrons, do not
affect the unperturbed energy. It is therefore necessary for
us to construct the secular equation for all these possible func
tions in order to find the correct combinations and first approxi
mation to the energy levels. 1
Before doing this, however, we should ask if there are any more
unperturbed wave functions belonging to this level. If, in setting
up the perturbation problem, we had called the term 2e 2 /7\/
the perturbation, then the singleelectron functions would have
been hydrogenlike functions with quantum numbers n, l y mi,
and m 8 . The energy of these solutions depends only on n,
as we have seen. However, a better starting point is to add and
subtract a term 2j)(xi) representing approximately the average
i
effect of the electrons on each other. If this term is added to
H° and subtracted from H', the true Hamiltonian H = H° + H'
is of course unaltered and the unperturbed equation is still
separable. The singleelectron functions are, however, no longer
hydrogenlike functions and their energies are no longer inde
pendent of the quantum number I, because it is only with a
Coulomb field that such a degeneracy exists (see Sec. 29a).
Therefore, in considering the wave functions to be combined we
do not ordinarily include any but those involving a single set of
values of n and l\ i.e., those belonging to a single configuration.
The consideration of a simple example, the configuration
ls 2 2p of lithium, may make clearer what the different unper
turbed functions are. Table 301 gives the sets of quantum num
1 The treatment of atoms which we are giving is due to J. C. Slater, Phys.
Rev. 34, 1293 (1929), who showed that this method was very much simpler
and more powerful than the complicated grouptheory methods previously
used.
234
MANYELECTRON ATOMS
[IXSOb
bers possible for this configuration. The notation (1003^)
means n = 1, I = 0, mi = 0, m, = +K Each line of the table
n corresponds to a set of functions u a • • • u v which when
substituted into the determinant of Equation 307 gives a satis
factory antisymmetrical wave function ^J corresponding to the
Table 301. — Sets of Quantum Numbers for the Configuration ls 2 2p
1. (100K) (ioo M) (2iiH);
2. (100K) (100 K) (211 K);
3. (lOOK) (ioo H) (210K);
4. (iooh) (ioo K) (210 H);
5. (100K) (100 K) (21 IK);
6. (iook) (ioo K) (2i l K);
2mi = +1, 2m, = +K,
Smz = +1, 2m, = K,
Smj = 0, 2m, = +K>
2w/ = 0, 2m, = K,
2m z = 1, 2m, = +K,
2mj = — 1, 2m, = — K
same unperturbed energy level. No other sets satisfying the
Pauli exclusion principle can be written for this configura
tion. The order of the expressions n, I, m t , m 8 in a given row is
unimportant.
This simple case illustrates the idea of completed shells of
electrons. The first two sets of quantum numbers remain the
Table 302. — Sets of Quantum Numbers for the Configuration np 2
2mj
2m,
1. (nllK)
(nil K)
2
2. (nliy 2 )
(nlOK)
1
+ 1
3. (nllK)
(mo y 2 )
1
4. (nil ~K)
(nlOK)
1
5. (nil ~K)
(nlO K)
1
1
6. (nllH)
(nl IK)
+1
7. (nliy 2 )
(nllK)
8. (nil ~K)
(nl IK)
9. (nlOK)
(nlO K)
10. (nil K)
(nl 1 K)
1
11. (nl IK)
(nlOK)
1
+1
12. (nl 1 H) (nl0y 2 )
1
13. (nl IK)
(nlO K)
1
14. (nl 1 K) (nlO K)
1
1
15. (nl IK)
(nl l'K)
2
same throughout this table because Is 2 is a completed shell;
i.e., it contains as many electrons as there are possible sets of
quantum numbers. The shell ns can contain two electrons,
np six electrons, nd ten electrons, etc. In determining the
EX30C] SLATER'S TREATMENT OF COMPLEX ATOMS 235
number of wave functions which must be combined, it is only
necessary to consider electrons outside of completed shells,
because there can be only one set of functions Ua • • • u v i or the
completed shells.
Table 302 gives the allowed sets of quantum numbers for two
equivalent p electrons, i.e., two electrons with the same value
of n and with I = 1.
Problem 301. Construct tables similar to Table 302 for the configura
tions np 3 and nd 2 .
30c. Factorization and Solution of the Secular Equation. — We
have now determined the unperturbed wave functions which
must be combined in order to get the correct zerothorder wave
functions for the atom. The next step is to set up the secular
equation for these functions as required by perturbation theory,
the form given at the end of Section 24 being the most con
venient. This equation has the form
#n  W H l2 • • • H lk
#21 #22 — W * H 2 k
Hki Hk2 ' ' ' Hkk — W
= 0, (308)
in which
H nm = ftfH+mdr. (309)
\p n is an antisymmetric normalized wave function of the form of
Equation 306 or 307, the functions u composing it correspond
ing to the nth row of a table such as Table 301 or 302. H is
the true Hamiltonian for the atom, including the interactions
of the electrons.
This equation is of the fcth degree, k being the number of
allowed sets of functions u a • • • u v . Thus for the configura
tion ls 2 2p k is equal to 6, as is seen from Table 301. However,
there is a theorem which greatly simplifies the solution of this
equation: the integral H mn is zero unless \{/ m and \l/ n have the same
value of Sm, and the same value of 2m i, these quantities being
the sums of quantum numbers m 8 and mi of the functions u making
up \p m and \l/ n . We shall prove this theorem in Section 30d in
connection with the evaluation of the integrals H mn} and in the
meantime we shall employ the result to factor the secular
equation.
236
MANYELECTRON ATOMS
[IX30c
Examining Table 301, we see that the secular equation for
ls 2 2p factors into six linear factors; i.e., no two functions yj/ n
and ^» have the same values of Sm 8 and Sm^. The equation
for np 2 , as seen from Table 302, has the factors indicated by
Figure 301, the shaded squares being the only nonzero elements.
A fifteenthdegree equation has, therefore, by the use of this
theorem been reduced to a cubic, two quadratic, and eight
linear factors.
Fig. 301. — The Becular determinant for the configuration np 2 , represented
diagrammatically.
By evaluating the integrals H mn and solving these equations,
the approximate energy levels W corresponding to this con
figuration could be obtained; but a still simpler method is
available, based on the fact that the roots W of the equations of
lower degree will coincide with some of the roots of the equations
of higher degree. The reason for this may be made clear by the
following argument. The wave . functions \pi, ^ 2 , • • • , 4>kj
which we are combining, differ from one another only in the
quantum numbers m 8 and mi of the single electrons, these
quantum numbers representing the z components of the spin
and orbital angular momenta of the electrons. The energy
of a single electron in a central field does not depend on mi or m 8
(neglecting magnetic effects), since these quantum numbers
refer essentially to orientation in space. The energy of an
atom with several electrons does depend on these quantum
numbers, because the mutual interaction of the 'electrons is
influenced by the relative orientations of the angularmomentum
IX30c] SLATER'S TREATMENT OF COMPLEX ATOMS 237
vectors of the individual electrons. Just as for oneelectron
atoms, however, the orientation of the whole atom in space
does not affect its energy and we expect to find a number of states
having the same energy but corresponding to different values
of the z components of the total orbital angular momentum
and of the total spin angular momentum; i.e., to different values
of 2m i and 2m„
This type of argument is the basis of the vector model 1 for atoms,
a very convenient method of illustrating and remembering the
results of quantummechanical discussions such as the one we
are giving here. In the vector model of the atom the orbital
and spin angular momenta of the individual electrons are con
sidered as vectors (see Section le) which may be combined to
give resultant vectors for the whole atom, the manner in which
these vectors are allowed to combine being restricted by certain
rules in such a way as to duplicate the results of quantum
mechanics. The vector picture is especially useful in classifying
and naming the energy levels of an atom, the values of the
resultant vectors being used to specify the different levels.
In Chapter XV we shall show that not only is the energy of a
stationary state of a free atom a quantity which has a definite
value (and not a probability distribution of values) but also the
total angular momentum and the component of angular momen
tum in any one chosen direction (say the z direction) are similar
quantities. Whereas it is not possible to specify exactly both the
energy and the positions of the electrons in an atom, it is possible
to specify the above three quantities simultaneously. If the
magnetic effects are neglected we may go further and specify the
total spin and total orbital angular momenta separately, and
likewise their z components. However, we may not give
the angular momenta of the individual electrons separately,
these being quantities which fluctuate because of the electron
interactions.
It will likewise be shown that when magnetic effects are
neglected the square of the total orbital angular momentum
must assume only the quantized values L(L + l)(h/2w) 2 where
L is an integer, while the square of the total spin angular momen
tum can take on only the values S(S + 1)(A/2t) 2 where S is
integral or halfintegral. (The letter L is usually used for the
1 See Pauling and Gotjdsmit, "The Structure of Line Spectra."
238 MANYELECTRON ATOMS [IX30c
total resultant orbital angular momentum of the atom, and the
letter S for the total spin angular momentum; see Section 296.)
In the approximation 1 we are using, the states of an atom may
be labeled by giving the configuration and the quantum numbers
L, S, Ml = 2m*, and Ms = 2ra„ the last two having no effect
on the energy. Just as for one electron, the allowed values
of Ml are L, L — 1, • • • , — L + 1, — L] M s is similarly
restricted to S, S — 1, • • • , —8 + 1, — S, all of these values
of Ms and M L belonging to the same degenerate energy level and
corresponding to different orientations in space of the vectors
L and S.
We shall now apply these ideas to the solution of the secular
equation, taking the configuration np 2 as an example. From
Table 302 we see that Hu — W is a linear factor of the equation,
since ^i alone has 2mi = 2 and Xm 8 = 0. A state with Ml = 2
must from the above considerations have L ^ 2. Since 2 is
the highest value of Ml in the table, it must correspond to
L = 2. Furthermore the state must have S = 0, because
otherwise there would appear entries in the table with Ml = 2
and Ms ^ 0. This same root W must appear five times in the
secular equation, corresponding to the degenerate states L = 2,
S = 0, M s = 0, Ml = 2, 1, 0, 1, 2. From this it is seen
that this root (which can be obtained from the linear factors)
must occur in two of the linear factors (Ml = 2,2; Ms = 0),
in two of the quadratic factors (Ml = 1,1; Ms = 0), and in
the cubic factor (Ml = 0, Ms = 0). The linear factor H 2 2 — W
with Ml = 1, Ms = 1 must belong to the level L = 1, S = 1,
because no terms with higher values of M L and Ms appear in
the table except those already accounted for. This level will
correspond to the nine states with Ml = 1,0, — 1, and Ms = 1,
0, — 1. Six of these are roots of linear factors (M L = ±1,
Ms = ±1; Ml = 0; M a = ±1), two of them are roots of the
quadratic factors (Ml = ±1, Ms = 0), and one is a root of the
cubic factor (M L = 0, Ms = 0).
Without actually solving the quadratic equations or evaluating
the integrals involved in them, we have determined their roots,
since all the roots of the quadratics occur also in linear factors.
1 This approximation, called (LS) or RussellSaunders coupling, is valid
for light atoms. Other approximations must be made for heavy atoms in
which the magnetic effects are more important.
EMOd] SLATER'S TREATMENT OF COMPLEX ATOMS 239
Likewise we have obtained two of the three roots of the cubic.
The third root of the cubic can be evaluated without solving the
cubic or calculating the nondiagonal elements of the equation,
by appealing to the theorem that the sum of the roots of a
secular equation (or of one of its factors) is equal to the sum
of the diagonal elements of the equation 1 (or of the factor).
Since two of the roots of the cubic have been found and the sum
of the roots is given by the theorem, the third may be found.
It corresponds to the state L = 0, S = 0, since this is the only
possibility left giving one state with Ml = and Ms = 0.
The three energy levels for np 2 which we have found are
W = # u , l D (L = 2, S = 0, M 8  0, Ml = 2, 1, 0, 1, 2);\
W = #22, 3 P (L = 1, S = 1, Ms = ±1, 0; Ml = ±1, 0);(
W = #77 + #88 + #99  #11  #22, *S (L = 0, S = 0,
Ms = 0, Ml = 0).
(3010)
The term symbols 1 D, 3 P, *S have been explained in Section 296.
Problem 302. Investigate the factorization of the secular equation
for np 3 , using the results of Problem 301, and list terms which belong to
this configuration.
30d. Evaluation of Integrals. — We need to obtain expressions
for integrals of the type
# mn = j+*H+ n dr = ^22 ( "" l)p+p 'J p,w * (l) • • •
p p'
u*(N)HPu a (l) • • • u,(N)dr. (3011)
1 To prove this theorem, we expand the secular equation 308 and arrange
according to powers of W. The resulting algebraic equation in W will have
k roots, TTi, W2, • • • , Wk and can therefore be factored into k factors
(W  WMW  W 2) • • • (W  W k ) = 0.
The coefficient of — W k ~ l in this form of the equation is seen to be
Wi + W 2 + • • • + W k ;
the coefficient of  W k ~ l in Equation 308 is seen to be
Hw + H22 + • • • ■+■ Hkk.
These two expressions must therefore be equal, which proves the theorem.
240 MANYELECTRON ATOMS [KS0d
We may eliminate one of the summations by the following
device:
JP f u*(l) • • • u*(N)HPu a (l) • • • u v {N)dr =
P"JP'u*(l) • • • u*(N)HPu a (l) • • • !*,(#)*• =
JP"P'u*(l) • • • u*(N)HP"Pu a (\) • • • !*,(#)*.
the first step being allowed because P" only interchanges the
names of the variables of a definite integral. If we choose P"
to be P' 1 , the inverse permutation to P' 9 then P"P f = 1; i.e.,
P'T' is the identity operation, while P"P is still some member
of the set of permutations, all members of which are summed
over. The integral therefore no longer involves P f and the
sum over P' reduces to multiplication byiV!, the number of
permutations. We thus obtain the equation
H mn = 2(l) p K*(D • • • u*(N)HPu a (l) • •  u v {N)dr.
p
(3012)
We shall now prove the theorem that H mn = unless 2m, is
the same for ^ m and ^ n . i? does not involve the spin coordinates
so that integration over these coordinates yields a product of
orthogonality integrals for the spin functions of the various
electrons. Unless the spins of corresponding electrons in the
two functions u*(l) • • • u*(N) and Pu a (l) • • • u v (N) are
the same, the integral is zero. If 2m, is not the same for \l/ m
and ^ n there can be no permutation P which will make such a
matching of the spins possible, because the number of positive
and negative spins is different in the two functions.
To prove the theorem concerning Sraj it is necessary to specify
further the nature of H. We write
H = %f t + Xf/a,
«j>«
where
fi  &^r„ v?  77 and 9ii = ui
The functions w«(l) • • • are solutions of
( h 2 %e 2 )
EMOd] SLATER'S TREATMENT OF COMPLEX ATOMS 241
From this we see that
fiU&) =/(r<)w f (i),
where /(r») is a function of r< alone. The integral of the first
term in H thus reduces to
X(l) r %$<(\)Pu a {l)dn ■ ■ ■ fu?(i)f(ri)Pu t (i)dn ■ ■ •
P I
Su?{N)Pu v (N)dr N , (3013)
in which Pu^ (i) is used as a symbol for u r (j) in which electron j
has replaced i as a result of the permutation P. Because of the
orthogonality of the u'&, this is zero unless Pu^(i) = u^(j) except
perhaps for j equal to the one value i. In addition, since
uS) = Rm(Ti) • e lm (#i) • e im % (3014)
the factor fu?(i)f(ri)Pu{(i)dTi will be zero unless u f (i) and
Puf(i) have the same quantum number m t . We thus see that
this integral will vanish, unless all the u's but one pair match and
the members of that pair have the same value of mi.
Similar treatment of the term Xg^ shows that all but perhaps
two pairs must match. The factor containing these unmatched
functions is
M(i)ut(j)fPut(i)Pui(j)dT4Ti. (3015)
It can be shown 1 that
k ^5^4*^^*^^*'^^' (30_16)
k,tn
in which r a is the smaller of r< and r i} and r b is the greater.
PJ^cos #) is an associated Legendre function, discussed in
Section 196. Using this expansion we obtain for the <p part of
the above integral f *f r e i{Pm i^i^ ) ^e i{Pm 'i^i^ ) ^d(p % d(p h in which
mi is associated with u^(i) f m\ with u^j), Pmi with Pu{(i), and
Pm\ with Put(j). This vanishes unless Pmi — mi + m =
and Pmi — m\ — m = 0; i.e., unless Pmi + Pm\ = mi + m\.
1 For a proof of this see J. H. Jeans, "Electricity and Magnetism," 5th ed„
Equations 152 and 196, Cambridge University Press, 1927.
242 MANYELECTRON ATOMS [IX30d
This completes the proof of the theorem that H mn = unless
Xmi is the same for \f/ n and ^ m .
Of the nonvanishing elements H mn only certain of the diagonal
ones need to be evaluated in order to calculate the energy levels,
as we have seen in the last section. Because of the orthogonality
of the u'a, Equation 3013 vanishes unless P = 1 (the identity
operation) when a diagonal element H mm is being considered.
Since the u y s are also normalized, this expression reduces to
%futti)f(ri)u t (i)d Ti = J/,, (3017)
t X
a relation which defines the quantities I,.
Similarly, the orthogonality of the u's restricts P in Equation
3015 to P = 1 and P = (ij), the identity operation and the
interchange of i and j, respectively. The first choice of P
contributes the terms
2 Jti r *(t>f0^W«f0')*4r / ~ 2« 7 <» (3018)
while the second yields
 2 J ^W^O")^ (t>r0>*n*/ = ~ 2*"' (3 °~ 19)
»J>* *J>»
The integral if <; vanishes unless the spins of w f (i) and i^O') are
parallel, i.e., unless m.< = m 8] .
The functions /* reduce to integrals over the radial part of
u(i),
h = JRtiiriWdRniirddu. (3020)
We shall not evaluate these further.
The functions J», and Ka may" be evaluated by using the
expansion for 1/nj given in Equation 3016. For J a the (pi
part of the integral has the form f T e im *id<p iy which vanishes
unless m = 0. The double sum in the expansion 3016 thus
reduces to a single sum over k, which can be written
J a = %a k (lmi; Vm[)F k (nl; n'V), (3021)
k
in which nlmi and n'Vm\ are the quantum numbers previously
represented by n< and n y , respectively. a k and F* are given by
IX30d] SLATER'S TREATMENT OF COMPLEX ATOMS 243
aHlmrl'm')  (2 * ± 1)( *  m ' l)! (2Z ' ± M Z ND'
a (fcm, £ m,)  2(z + m ^ f 2 (J' + mf)!
jT T {P^'(cos t ) ) 2 P*°(cos #»•) sin #«Z0,
jT x {P^'i(cos #,) } 2 P°(cos *,) sin tfyto/, (3022)
and
F*(nl; nT) = (^) 2 e 2 Jj " J[ ^S/^O^^rCry)^?^^,, (3023)
The a's are obtained from the angular parts of the wave functions,
which are the same as for the hydrogen atom (Tables 211 and
212, Chap. V). Some of these are given in Table 303, taken
Table 303. — Values of a k (lmi; I'm'i)
(In cases with two ± signs, the two can be combined in any of the four
possible ways)
Electrons
I
rrt\
V
mj
a°
a 2
a*
88
1
sp
1
±1
1
VV
1
±1
1
±1
K 5
1
±1
1
%5
1
1
^5
8d
2
±2
2
±1
2
pd
±1
2
±2
%5
±1
2
±1
Ms
±1
2
«5
2
±2
"«6
2
±1
^5
2
^5
dd
2
±2
2
±2
**9
J*41
2
±2
2
±1
%>
«41
2
±2
2
«9
%41
2
±1
2
±1
K9
J %41
2
±1
2
K9
 2 **41
2
2
Yt*
8 $441
244 MANYELECTRON ATOMS [IX80e
from Slater's paper. The P's, on the other hand, depend on the
radial parts of the wave functions, which for the best approxima
tion are not hydrogenlike.
Kij may be similarly expressed as
Ku = Xb\lmi) I'mWinl; nT), (3(H2 4)
k
in which
b k {lmi)Vm[) =
(t  \mi  m{)l(2i + l)(l  h^KgT + l)(i'  \m[\) \
4(t + m, mj)!(l + m«)(r + m{)!
j Ppj^cos ^P^ 1 (cos tf)PJprtf (cos #) sin &»  *, (3025)
and
G k (nl;n'V) = e 2 (4ir)*J[ <D j[ <D 2J»,(n)B»^(r < )B»i(ry)12»*r(r / )
^rtfdr«dr,. (3026)
The functions 6* are given in Table 304. The functions G k
are characteristic of the atom.
30e. Empirical Evaluation of Integrals. Applications. — We
have now carried the computations to a stage at which the
energy levels may be expressed in terms of certain integrals
Ii, F k ) and G k which involve the radial factors of the wave
functions. One method of proceeding further would be to assume
some form for the central field fl(r t ), determine the functions
Rmi(ri), and use them to evaluate the integrals. However,
another and simpler method is available for testing the validity
of this approximation, consisting in the use of the empirically
determined energy levels to evaluate the integrals, a check on
the theory resulting from the fact that there are more known
energy levels than integrals to be determined.
For example, if we substitute for ff n, etc., for the configuration
np 2 the expression in terms of J t , F k , and G k , using the results of
the previous section and Equation 3010, we obtain for the
energies of the terms 1 Z), 3 P, and X S the quantities
>D:TF = 2/(n, 1) + F« + % 5 F\
V: W = 2/(n, 1) + F°  Y 2b F*  % 6 Q*,
*S: W = 2/(n, 1) + F° + W + %5<? 2 
IX30e] SLATER'S TREATMENT OF COMPLEX ATOMS
245
Table 304. — Values of 6*(Zmj; j'mj)
(In cases where there are two ± signs, the two upper, or the two lower,
signs must be taken together)
Electrons
I
mi
i
m'j
b Q
6 1
6*
6 s
b*
88
1
8p
i
±1
X
i
X
VV
1
±1
i
±1
1
«5
1
±1
i
K 5
1
±1
i
+ 1
^5
1
i
1
^5
sd
2
±2
X
2
±1
X
2
X
pd
±1
2
±2
X
%45
±1
2
±1
h
%45
±1
2
X*
x x«
±1
2
+ 1
3 %45
±1
2
+ 2
4 ^45
2
±2
^45
2
±1
X
2 ^45
2
Vl5
2 %45
dd
2
±2
2
±2
1
H9
Ha
2
±2
2
±1
%9
%41
2
±2
2
n*
15 A*i
2
±2
2
Tl
35 441
2
±2
2
=F2
7 ^41
2
±1
2
±1
1
X9
*9i41
2
±1
2
X9
*%4l
2
±1
2
Tl
%9
4 %41
2
2
1
%9
3 %41
Examination of Equations 3018 and 3019 shows that for
equivalent electrons F is equal to G (with the same index). We
therefore have for the separations of the levels for np 2
iD *P =% 5 F*(nl;nl),
ifl iD = ^ 5 F 2 (nl;nl).
The theory therefore indicates that, if the approximations which
246 MANYELECTRON ATOMS [IX31
have been made are valid, the ratio of these intervals should be
2:3, a result which is obtained without the evaluation of any
radial integrals at all. In addition, since F 2 is necessarily
positive, this theory gives the order of the terms, 3 P lying lowest,
l D next, and l S highest. This result is in agreement with
Hund's empirical rules, that terms with largest multiplicity
usually lie lowest, and that, for a given multiplicity, terms
with largest L values usually lie lowest. 1
Slater gives the example of the configuration 2 ls 2 2s 2 2p 6
3s 2 3p 2 of silicon, for which the observed term values 3 are
3 P = 65615 cm" 1 ,
l D = 59466 cm" 1 ,
l S = 50370 cm" 1 ,
so that the ratio l D  3 P to l S  l D is 2:2.96, in excellent
agreement with the theory. In other applications, however,
large deviations have been found, most of which have been
explained by considering higher approximations based on the
same general principles. 4
31. VARIATION TREATMENTS FOR SIMPLE ATOMS
The general discussion of Section 30, which is essentially a
perturbation calculation, is not capable of very high accuracy,
especially since it is not ordinarily practicable to utilize any
central field except the coulombic one leading to hydrogenlike
orbital functions. In this section we shall consider the applica
tion of the variation method (Sec. 26) to lowlying states
of simple atoms such as lithium and beryllium. This type of
treatment is much more limited than that of the previous
section, but for the few states of simple atoms to which it has
been applied it is more accurate.
1 Pauling and Goudsmit, "The Structure of Line Spectra," p. 166.
2 This configuration gives the same interval ratios as np 2 } only the absolute
energy being changed by the presence of the closed shells.
3 As mentioned in Section 5a, term values are usually given in cm 1 and
are measured downward from the lowest state of the ionized atom. Hence
the largest term value represents the lowest energy level.
4 There have been many papers on this subject; a few are: C. W. Ufford,
Phys. Rev. 44, 732 (1933); G. H. Shortley, Phys. Rev. 43, 451 (1933); M. H.
Johnson, Jr., Phys. Rev. 43, 632 (1933); D. R. Inglis and N. Ginsburg,
Phys. Rev. 43, 194 (1933). A thorough treatment is given by E. U. Condon
and G. H. Shortlev, "The Theory of Atomic Spectra," Cambridge, 1935.
IX31a] VARIATION TREATMENTS FOR SIMPLE ATOMS 247
The principles involved are exactly the same as those discussed
in Section 26 and applied to helium in Section 29c, so we shall
not discuss them further but instead study the different types of
variation functions used and the results achieved.
31a. The Lithium Atom and Three electron Ions. — Table 311
lists the variation functions which have been tried for the
lowest state of lithium, which has the configuration ls 2 2s. All
these functions are of the determinant type given in Equation
307 and in all of them the orbital part of u u (i) is of the form
e ao , in which Z', the effective atomic number for the K shell,
is one of the parameters determined by the variation method.
The table gives the expressions for 6, the orbital part of u 2a (i),
the function for the 2s electron. In addition, the upper limit
to the total energy of the atom is given, and also the value of the
first ionization potential calculated by subtracting the value of
the energy calculated for Li + from the total energy calculated
for Li. The Li + calculation was made with the use of the same
type of Is function used in Li for the K shell, in order to cancel
part of the error introduced by this rather poor K function.
The table also gives the differences between these calculated
quantities and the experimental values.
Table 311. — Variation Functions for the Normal Lithium Atom
Units: RJw
Experimental total energy: —14.9674; experimental ionization potential:
0.3966
2s function 1
Total
energy
Differ
ence
Ionization
potential
Differ
ence
1.
r
14.7844
0.1830
0.3392
0.0574
2.
b = re ^
14.8358
.1316
.3906
.0060
3.
b = e ~%i 1 )
14.8366
.1308
.3912
.0054
4.
r  rr—  f —
b  ae O0  e ao
ao
14.8384
.1290
.3930
.0036
i The function 1 was used by C. Eckart, Pkya. Rev. 36, 878 (1930), 2 and 3 by V. G ille
min and C. Zener, Z. /. Phya. 61, 199 (1930), and 4 by E. B. Wilson, Jr., /. Chem. Phya. 1,
211 (1933). The last paper includes similar tables for the ions Be + , B ++ , and C +++ .
248
MANYELECTRON ATOMS
[IX31a
Table 312 lists the best values of the parameters for these
lithium variation functions. Figure 311 shows the total
electron distribution function 4xr 2 p for lithium, calculated using
Table 312. — Parameter Values for Lithium Variation Functions
Z'r
Mli
= e ao
Function
Z'
V
.
r
1
b = e Hv 1 )
2 686
0.888
\ do /
2.
r
b = re ^
2.688
.630
3.
b = e °°( a. 1 )
2 . 688
.630
5 56
4.
b = ore <*  e ao . ..
2.69
.665
1.34
1.5
the best of these functions, p is the electron density, which can
be calculated from f in the following manner:
P = 3JW*idr*.
(311)
\p*\f/dTidT 2 dn gives the probability of finding electron 1 in the
volume element dri, electron 2 in dr 2 , and electron 3 in dr z .
Integration over the coordinates of electrons 1 and 2 gives the
probability of finding electron 3 in dr 3 , regardless of the positions
of 1 and 2. Since \f/*\p is symmetric in the three electrons, the
probability of finding one electron in a volume element dxdydz in
ordinary threedimensional space is three times the probability of
finding a particular one. Figure 311 shows clearly the two shells
of electrons in lithium, the wellmarked K shell and the more
diffuse L shell. Due to the equivalence of the three electrons,
we cannot say that a certain two occupy the K shell and the
remaining one the L shell, but we can say that on the average
there are two electrons in the K shell and one in the L shell.
The next step to be taken is to apply a variation function to
lithium which recognizes explicitly the instantaneous, instead
of just the average, influence of the electrons on each other.
Such functions were found necessary to secure really accurate
results for helium (Sec. 29c), but their application to lithium
IX81b] VARIATION TREATMENTS FOR SIMPLE ATOMS 249
involves extremely great complications. This work has been
begun (by James and Coolidge at Harvard 1 ).
31b. Variation Treatments of Other Atoms. — Few efforts have
been made to treat more complicated atoms by this method.
Beryllium has been studied by several investigators but the
functions which give good results for lithium are not nearly so
accurate for heavier atoms. Hydrogenlike functions with
variable effective nuclear charges (function 1 of Table 311 is
Fig. 311. — The electron distribution function D = 4xr 2 p for the normal lithium
atom.
such a function for n = 2, I = 0) have been applied to the case
of the carbon atom, 2 the results being in approximate agreement
with experiment. Functions of the types 2 and 3 of Table 311
have also been tried 3 for Be, B, C, N, 0, F, and Ne. A more
satisfactory attack has been begun by Morse and Young, 4 who
have prepared numerical tables of integrals for wave functions
dependent on four parameters (one for Is, two for 2s, and one
1 Private communication to the authors; see H. M. James and A. S.
Coolidge, Phys. Rev. 47, 700 (1935), for a preliminary report.
1 N. F. Bbakdslby, Phys. Rev. 39, 913 (1932).
8 C. Zbner, Phys. Rev. 36, 51 (1930).
4 P. M. Morse and L. A. Young, unpublished calculations (available av
the Massachusetts Institute of Technology).
250 MANYELECTRON ATOMS [IX32a
for 2p oneelectron functions) for the treatment of the K and L
shells of atoms.
The analytical treatment of complicated atoms by this method
is at present too laborious for the accuracy obtained, but it may
be possible to find new forms for the variation function which will
enable further progress to be made.
32. THE METHOD OF THE SELFCONSISTENT FIELD
The previous sections give some indication of the difficulty
of treating manyelectron atoms in even an approximate manner.
In this section we shall discuss what is probably the most success
ful effort which has yet been made in attacking this problem,
at least for those atoms which are too complicated to treat by
any satisfactory variation function. Both the principle and
the difficult technique involved are due to Hartree, 1 who, with the
aid of his students, has now made the numerical computations
for a number of atoms. In Section 326 we shall show the
connection between this method and those previously discussed.
32a. Principle of the Method. — In Section 306 we have
pointed out that the wave equation for a manyelectron atom
can be separated into singleelectron wave functions not only
when the mutual interactions of the electrons are completely
neglected but also when a central field v(x t ) for each electron is
added to the unperturbed equation and subtracted from the
perturbation term. Each of the resulting separated unper
turbed wave equations describes the motion of an electron in a
central field which is independent of the coordinates of the
other electrons. The perturbation treatment considered in
Section 30 was based on the idea that a suitable choice could be
made of these central fields for the individual electrons so that
they would represent as closely as possible the average effect
upon one electron of all the other electrons in the atom.
The important step in the application of such a method of
treatment is the choice of the potentialenergy functions repre
senting the central fields. The assumption made by Hartree
is that the potentialenergy function for one electron due to a
second electron is determined approximately by the wave
function for the second electron, m^(2), say, being given by the
1 D. R. Hartree, Proc. Cambridge Phil Soc. 24, 89, 111, 426 (1928).
IX32a] THE METHOD OF THE SELFCONSISTENT FIELD 251
potential corresponding to the distribution of electricity deter
mined by the probability distribution function u$(2) u fi (2).
This is equivalent to assuming that the wave function for the
second electron is independent of the coordinates of the first
electron. The complete centralfield potentialenergy function
for the first electron is then obtained by adding to the potential
energy function due to the nucleus those potentialenergy
functions due to all the other electrons, calculated in the way
just described. The wave function for the first electron can
then be found by solving the wave equation containing this
complete potentialenergy function.
It is seen, however, that in formulating a method of calculating
the functions u^(k) for an atom we have assumed them to be
known. In practice there is adopted a method of successive
approximations, each cycle of which involves the following
steps :
1. A potentialenergy function due to the nucleus and all of
the electrons is estimated.
2. From this there is subtracted the estimated contribution
of the fcth electron, leaving the effective potentialenergy function
for this electron.
3. The resulting wave equation for the fcth electron is then
solved, to give the wave function u^(k). Steps 2 and 3 are
carried out for all of the electrons in the atom.
4. Using the functions u^(k) obtained by step 3, the potential
energy functions due to the various electrons are calculated,
and compared with those initially assumed in steps 1 and 2.
In general the final potentialenergy functions are not identical
with those chosen initially. The cycle is then repeated, using
the results of step 4 as an aid in the estimation of new potential
energy functions. Ultimately a cycle may be carried through
for which the final potentialenergy functions are identical
(to within the desired accuracy) with the initial ones. The
field corresponding to this cycle is called a selfconsistent field
for the atom.
It may be mentioned that the potentialenergy function due
to an $ electron is spherically symmetrical, inasmuch as the
probability distribution function u^u na is independent of <p and #.
Moreover, as a result of the theorem of Equation 2116 the
potentialenergy function due to a completed shell of electrons
252 MANYELECTRON ATOMS [IX82b
is also spherically symmetrical. Spherical symmetry of the
potential function greatly increases the ease of solution of the
wave equation.
Hartree employs the method of numerical integration sketched
in Section 27c to solve the singleelectron wave equations.
In addition he makes the approximation of considering all
contributions to the field as spherically symmetrical. Thus
if some electron (such as a p electron) gives rise to a charge
distribution which is not spherically symmetrical, this is averaged
over all directions. Finally, the simple product of Equation
302 is used for the wave function for the whole atom. As we
have seen, this does not have the correct symmetry required by
Pauli's principle. The error due to this involves the interchange
energies of the electrons (Sec. 32c).
32b. Relation of the Selfconsistent Field Method to the
Variation Principle. — If we choose a variation function of the
form
<t> = t*«(l) u p (2) • • • u y (N) (321)
and determine the functions u^(i) by varying them individually
until the variational integral in Equation 261 is a minimum,
then, as shown in Section 26a, these are the best forms for the
functions %(t) to use in a wave function of this product type for
the lowest state. Neglecting the fact that Hartree averages
all fields to make them spherically symmetrical, we shall now
show 1 that the variationprinciple criterion is identical, for this
type of <£, with the criterion of the selfconsistent field. If
we keep each u^(i) normalized, then f<t>*<t>d,T = 1 and
E = f<t>*H<txlT. (322)
The operator H may be written as •
H = 2 H < + 2$ (32  3)
with
*,;>t
h 2 Ze 2
Hi = «4v?  — ' (324)
87T 2 m Ti v
1 J. C. Slater, Phys. Rev. 35, 210 (1930); V. Fock, Z. f. Phys. 61, 126
(1930).
EC82b] THE METHOD OF THE SELFCONSISTENT FIELD 253
Using this and the expression for <t> in Equation 321, we obtain
\?(i)ut(j)^u { (i)u e (j)dT4Ti. (325)
2J/'
The variation principle can now be applied. This states that
the best form for any function u^(i) is the one which makes E
a minimum (keeping the function normalized). For this mini
mum, a small change 8u^(i) in the form of U[(i) will produce no
change in E\ that is 8E = 0.
The relation between 8u^(i) and 8E is
.JV
8E = 8\ u?(i)H&t(i)dTi +
sw
■?(*)*?(J)tr*r(i)«iU)dT4T t , (326)
in which the prime on the summation sign indicates that the
term with j = i is not included. Let us now introduce the new
symbol F if defined by the equation
F i = H i + ^ f t*f (jJ^WfO')*/, (327)
or
Fi = H { + V i9
in which
y
Fi is an effective Hamiltonian function for the ith electron, and
Vi the effective potentialenergy function for the ith electron
due to its interaction with the other electrons in the atom.
Using the symbol Fi, we obtain as the condition that E be sta
tionary with respect to variation in u^(i) the expression (Eq.
326)
8E = 6fu?(i)F<UtWn  0. (329)
254 MANYELECTRON ATOMS [IX32c
A similar condition holds for each of the N oneelectron functions
tta(l), • • • , u,(N).
Let us now examine the criterion used in the method of the
selfconsistent field. In this treatment the wave function
U[(i) is obtained as the solution of the wave equation
VfwrM + ^r(*i + ™  ^)%« = 0, (3210)
or, introducing the symbol F if
FMi) = e x Ui(i). (3211)
We know, however, that a normalized function u^(i) satisfying
this equation also satisfies the corresponding variational equation
h$u?(i)F x u ; {i)dT x = 0. (3212)
Equations 329 and 3212 are identical, so that by using the
variation method with a producttype variation function we
obtain the same singleelectron functions as by applying the
criterion of the selfconsistent field.
32c. Results of the Selfconsistent Field Method. — Hartree
and others have applied the method of the selfconsistent field
to a number of atoms and ions. In one series of papers 1 Hartree
has published tables of values of singleelectron wave functions
for Cl~, Cu + , K + , and Rb + . These wave functions, as given,
are not normalized or mutually orthogonal, but values of the
normalizing factors are reported. For these atoms the total
energy has not been calculated, although values of the individual
€t's are tabulated. (The sum of these is not equal to the total
energy, even if interchange is neglected.) For O, + , ++ , and
+++ , Hartree and Black 2 have given not only the wave functions
but also the total energies calculated by inserting these single
electron wave functions into a determinant such as Equation
307 and evaluating the integral E '= f\p*H\pdT.
Several other applications 3 have been made of this method and
a considerable number are now in progress. Slater 4 has taken
Hartree's results for certain atoms and has found analytic expres
1 D. R. Hartree, Proc. Roy. Soc. A 141, 282 (1933); A 143, 506 (1933).
2 D. R. Hartree and M. M. Black, Proc. Roy. Soc. A 139, 311 (1933).
3 F. W. Brown, Phys. Rev. 44, 214 (1933); F. W. Brown, J. H. Bartlett,
Jr., and C. G. Dunn, Phys. Rev. 44, 290 (1933); J. McDougall, Proc. Roy.
Soc. A 138, 550 (1932); C. C. Torrance, Phys. Rev. 46, 388 (1934).
* J. C. Slater, Phys. Rev. 42, 33 (1932).
IX32c] THE METHOD OF THE SELFCONSISTENT FIELD 255
sions for the singleelectron wave functions which fit these results
fairly accurately. Such functions are of course easier to use than
numerical data.
The most serious drawback to Hartree's method is probably
the neglect of interchange effects, i.e., the use of a simple product
Fxo. 321. — The electron distribution function D for the normal rubidium
atom, as calculated: I, by Hartree's method of the selfconsistent field; II, by the
screeningconstant method; and III, by the ThomasFermi statistical method.
type wave function instead of a properly antisymmetric one.
This error is partially eliminated by the procedure of Hartree
and Black described above, but, although in that way the energy
corresponding to a given set of functions %(fc) is properly calcu
lated, the functions %(&) themselves are not the best obtainable
because of the lack of antisymmetry of $. Fock 1 has considered
this question and has given equations which may be numerically
solved by methods similar to Hartree's, but which include inter
change. So far no applications have been made of these, but
several computations are in progress. 2
Figures 321, from Hartree, shows the electron distribution
function for Rb+ calculated by this method, together with those
given by other methods for comparison.
1 V. Fock, Z. f. Phys. 61, 126 (1930).
'See D. R. Hartree and W. Hartree, Proc. Roy. Soc. A 150, 9 (1936).
256 MANYELECTRON ATOMS [IX33a
Problem 321. (a) Obtain an expression for the potential due to an
electron in a hydrogenlike Is orbital with effective atomic number Z' 
21 A&' (&) Using this result, set up the wave equationfor one electron in a
helium atom in the field due to the nucleus and the other electron (assumed
to be represented by the wave function mentioned above). Solve the wave
equation by the method of difference equations (Sec. 27 d), and compare
the resultant wave function with that chosen initially.
33. OTHER METHODS FOR MANYELECTRON ATOMS
Besides the methods discussed in the previous sections there
are others yielding useful results, some of which will be briefly
outlined in the following sections. Several methods have been
proposed which are beyond the scope of this book, notably the
Dirac^Van Vleck 2 vector model, which yields results similar to
those given by the method of Slater of Section 30.
33a. Semiempirical Sets of Screening Constants. — One of the
methods mentioned in Section 316 consists in building up an
approximate wave function for an atom by the use of hydrogen
like singleelectron functions with effective nuclear charges
determined by the variation method. Instead of giving the
effective atomic number Z', it is convenient to use the difference
between the true atomic number and the effective atomic num
ber, this difference being called the screening constant. Pauling 3
has obtained sets of screening constants for all atoms, not by
the application of the variation method (which is too laborious),
but by several types of reasoning based in part on empirical
considerations, involving such quantities as xray term values
and molecular refraction values. It is not< to be expected
that wave functions formed in this manner will be of very great
accuracy, but for many purposes they are sufficient and for many
atoms they are the best available! The results obtained for
Rb + are shown in Figure 321.
Slater 4 has constructed a similar table, based, however, on
Zener's variationmethod calculations for the first ten elements
(Sec. 316). His screening constants are meant to be used in
1 P. A. M. Dirac, "The Principles of Quantum Mechanics," Chap. XI.
* J. H. Van Vleck, Phys. Rev. 46, 405 (1934).
8 L. Pauling, Proc. Roy. Soc. A 114, 181 (1927); L. Paulino and J.
Sherman, Z. f. Krist. 81, 1 (1932).
4 J. C. Slater, Phys. Rev. 36, 57 (1930).
IX33b] OTHER METHODS FOR MANYELECTRON ATOMS 257
functions of the type r n e~ z ' r instead of in hydrogenlike func
tions, the exponent n' being an effective quantum number.
A discussion of an approximate expression for the wave func
tion in the outer regions of atoms and ions and its use in the
treatment of various physical properties (polarizability, ioniza
tion potentials, ionic radii, etc.) has been given by Wasastjerna. 1
33b. The Thomas Fermi Statistical Atom. — In treating a
system containing a large number of particles statistical methods
are frequently applicable, so that it is natural to see if such
methods will give approximate results when applied to the
collection of electrons which surround the nucleus of a heavy
atom. Thomas 2 and Fermi 3 have published such a treatment.
In applying statistical mechanics to an electron cloud, it was
recognized that it is necessary to use the FermiDirac quantum
statistics, based on the Pauli exclusion principle, rather than
classical statistics, which is not even approximately correct for
an electron gas The distinctions between these have been men
tioned in Section 296 and will be further discussed in Section 49.
The statistical treatment of atoms yields electron distributions
that are surprisingly good in view of the small number of electrons
involved. These results have been widely used for calculating
the scattering power of an atom for zrays and for obtaining an
initial field for carrying out the selfconsistentfield computations
described in the previous section. However, the ThomasFermi
electron distribution does not show the finer features, such as
the concentration of the electrons into shells, which are character
istic of the more refined treatments. Figure 321 shows how the
ThomasFermi results compare with Hartree's and Pauling's
calculations for Rb+ .
General References on Line Spectra
Introductory treatments :
L. Pauling and S. Goudsmit: "The Structure of Line Spectra," McGraw
Hill Book Company, Inc., New York, 1930.
H. E. White: "Introduction to Atomic Spectra," McGrawHill Book
Company, Inc., New York, 1934.
1 J. A. Wasastjerna, Soc. Scieni. Fennica Comm. Phys.Math., vol. 6,
Numbers 1822 (1932).
*L. H. Thomas, Proc. Cambridge Phil. Soc. 23, 642 (1927).
3 E. Fermi, Z. /. Phys. 48, 73; 49, 560 (1928).
258 MANYELECTRON ATOMS [IX33b
A. E. Ruark and H. C. Urey: "Atoms, Molecules and Quanta,"
McGrawHill Book Company, Inc., New York, 1930.
A thorough quantummechanical treatment:
E. U. Condon and G. H. Shortley: "The Theory of Atomic Spectra,"
Cambridge University Press, 1935.
Tabulation of term values:
R. F. Bacher and S. Goudsmit: "Atomic Energy States," McGraw
Hill Book Company, Inc., New York, 1932.
CHAPTER X
THE ROTATION AND VIBRATION OF MOLECULES
The solution of the wave equation for any but the simplest
molecules (some of which are discussed in Chap. XII) is a very
difficult problem. However, the empirical results of molecular
spectroscopy show that in many cases the energy values bear a
simple relation to one another, such that the energy of the
molecule (aside from translational energy) can be conveniently
considered to be made up of several parts, called the electronic
energy, the vibrational energy, and the rotational energy.
This is indicated in Figure 341, showing some of the energy
levels for a molecule of carbon monoxide, as calculated from the
observed spectral lines by the Bohr frequency rule (Sec. 5a).
It is seen that the energy levels fall into widely separated
groups, which are said to correspond to different electronic states
of the molecule. For a given electronic state the levels are
again divided into groups, which follow one another at nearly
equal intervals. These are said to correspond to successive
states of vibration of the nuclei. Superimposed on this is the
fine structure due to the different states of rotation of the mole
cule, the successive rotational energy levels being separated by
larger and larger intervals with increasing rotational energy.
This simplicity of structure of the energy levels suggests that it
should be possible to devise a method of approximate solution
of the wave equation involving its separation into three equa
tions, one dealing with the motion of the electrons, one with
the vibrational motion of the nuclei, and one with the rotational
motion of the nuclei. A method of this character has been
developed and is discussed in the following section. The
remaining sections of this chapter are devoted to the detailed
treatment of the vibrational and rotational motion of molecules
of various types.
34. THE SEPARATION OF ELECTRONIC AND NUCLEAR MOTION
By making use of the fact that the mass of every atomic nucleus
is several thousand times as great as the mass of an electron,
259
260 THE ROTATION AND VIBRATION OF MOLECULES [X34
Born and Oppenheimer 1 were able to show that an approximate
solution of the complete wave equation for a molecule can be
obtained by first solving the wave equation for the electrons
cm"'
90,000
80,000
70,000

60,000
50,000
40,000
30,000

Etc.*
E+c.+
r
20,000
s~<
10,000
—
v>
Fig. 34rl. — Energy levels for the carbon monoxide molecule. On the left are
shown various electronic levels, with vibrational fine structure for the normal
state, and on the right, with one hundred fold increase of scale, the rotational
fine structure for the lowest vibrational level.
alone, with the nuclei in a fixed configuration, and then solving
a wave equation for the nuclei alone, in which a characteristic
energy value of the electronic wave equation, regarded as a
1 M. Bokn and J. R. Oppbnhbimbb, Ann. d. Phya. 84, 457 (1927).
X34] ELECTRONIC AND NUCLEAR MOTION 261
function of the internuclear distances, occurs as a potential
function. Even in its simplest form the argument of Born and
Oppenheimer is very long and complicated. On the other
hand, the results of their treatment can be very simply and
briefly described. Because of these facts, we shall content
ourselves with describing their conclusions in detail.
The complete wave equation for a molecule consisting of r
nuclei and s electrons is
r *
yi *i
in which M } is the mass of the jth nucleus, m the mass of each
electron, v ; ? the Laplace operator in terms of the coordinates
of the jth nucleus, and v» ? the same operator for the ith electron.
V is the potential energy of the system, of the form
the sums including each pair of particles once. Here Z, is the
atomic number of the jth nucleus.
Let us use the letter £ to represent the 3r coordinates of the
r nuclei, relative to axes fixed in space, and the letter x to repre
sent the 3s coordinates of the s electrons, relative to axes deter
mined by the coordinates of the nuclei (for example, as described
in Section 48). Let us also use the letter v to represent the
quantum numbers associated with the motion of the nuclei,
and n to represent those associated with the motion of the
electrons. The principal result of Born and Oppenheimer^
treatment is that an approximate solution ^«,„(x, £) of Equation
341 can be obtained of the form
4>nA*> f) = *n(*> £)*».,(£). (34r2)
The different functions \l/ n (x, £), which may be called the
electronic wave functions, correspond to different sets of values
of the electronic quantum numbers n only, being independent
of the nuclear quantum numbers v. On the other hand, each of
these functions is a function of the nuclear coordinates J as
well as the electronic coordinates x. These functions are
262 THE ROTATION AND VIBRATION OF MOLECULES [X84
obtained by solving a wave equation for the electrons alone, the
nuclei being restricted to a fixed configuration. This wave equation
is
8w 2 m {
2 *#«(*, «) + ^°{ V.(&  V{z, {)}*„<*, ) = 0. (343)
U(r)t
1
It is obtained from the complete wave equation 341 by omitting
the terms involving v/, replacing ^ by ^ n (x f £), and writing
U»(£) in place of W. The potential function V(x, £) is the
complete potential function of
Equation 341. It is seen that
for any fixed set of values of
the s nuclear coordinates £ this
equation 3^3, which we may
call the electronic wave equation,
is an ordinary wave equation for
the s electrons, the potential
energy function V being depend
ent on the values selected for the
nuclear coordinates £. In con
sequence the characteristic elec
Fig. 342.— A typical function u(r) for tronic energy values U n and the
a diatomic molecule (Morae function). , ,  , . .
electronic wave functions \j/ n
will also be dependent on the values selected for the nuclear
coordinates; we accordingly write them as (/»($) and yp n {x } £).
The first step in the treatment of a molecule is to solve this
electronic wave equation for all configurations of the nuclei.
It is found that the characteristic values U n (£) of the electronic
energy are continuous functions of the nuclear coordinates £.
For example, for a free diatomic molecule the electronic energy
function for the most stable electronic state (n = 0) is a function
only of the distance r between the two nuclei, and it is a con
tinuous function of r, such as shown in Figure 342.
Having evaluated the characteristic electronic energy £/ n (£)
as a function of the nuclear coordinates £ for a given set of
values of the electronic quantum numbers n by solving the
wave equation 343 for various nuclear configurations, we next
obtain expressions for the nuclear wave functions ^ n ,*({). It
was shown by Born and Oppenheimer that these functions are
X35] VIBRATION OF DIATOMIC MOLECULES 263
the acceptable solutions of a wave equation in the nuclear
coordinates £ in which the characteristic electronic energy
function U n (£) plays the role of the potential energy; that is,
the nuclear wave equation is
r
2i V ' ¥n '" a) + 1? {W "" ~ u »W+».>ti) = 0. (341)
yi
There is one such equation for each set of values of the electronic
quantum numbers n, and each of these equations possesses an
extensive set of solutions, corresponding to the allowed values
of the nuclear quantum numbers v. The values of W n , v are the
characteristic energy values for the entire molecule; they depend
on the electronic and nuclear quantum numbers n and v.
The foregoing treatment can be formally justified by a pro
cedure involving the expansion of the wave functions and
other quantities entering in the complete wave equation 341
as power series in (m /M) H , in which M is an average nuclear
mass. The physical argument supporting the treatment is
that on account of the disparity of masses of electrons and nuclei
the electrons carry out many cycles of their motion in the time
required for the nuclear configuration to change appreciably,
and that in consequence we are allowed to quantize their motion
for fixed configurations (by solving the electronic wave equation),
and then to use the electronic energy functions as potential energy
functions determining the motion of the nuclei.
When great accuracy is desired, and in certain cases when
only ordinary accuracy is required, it is necessary to consider the
coupling between electronic and nuclear motions, and especially
between the electronic angular momentum (either spin or
orbital) and the rotation of the molecule. We shall not discuss
these questions, 1 but shall treat only the simplest problems in
the complex field of molecular structure and molecular spectra
in the following sections. Some further discussion is also
given in Chapter XII and in Section 48 of Chapter XIV.
35. THE ROTATION AND VIBRATION OF DIATOMIC MOLECULES
In the previous section we have stated that an approximate
wave function for a molecule can be written as a product of two
1 See the references at the end of the chapter.
264 THE ROTATION AND VIBRATION OF MOLECULES [X36a
factors, one a function of the electronic coordinates relative to
the nuclei and the other a function of the nuclear coordinates.
In this section we shall consider the nuclear function and the
corresponding energy levels for the simplest case, the diatomic
molecule, assuming the electronic energy function U n (r) to be
known.
35a. The Separation of Variables and Solution of the Angular
Equations. — The wave equation for the rotation and vibration
of a diatomic molecule (Eq. 344) has the form
1 2 i i 1 9 i i °7T
V?*n,„ + irVty** + ^{Wn,,  U n (r))* n , v = 0, (351)
in which ^ n ,v= yfrn, v {x\, y\ y Zi, x 2y 2/2, £2) is the wave function for
the nuclear motion, M\ and M 2 are the masses of the two nuclei,
and
x ly y xy and z x being the Cartesian coordinates of the tth nucleus
relative to axes fixed in space. Equation 351 is identical with
the wave equation for the hydrogen atom, the two particles
here being the two nuclei instead of an electron and a proton.
We may therefore refer to the treatment which has already
been given of this equation in connection with hydrogen. All
the steps are the same until the form for U n (r) is inserted into
the radial equation.
In Section 18a we have seen that Equation 351, expressed in
terms of the Cartesian coordinates of the two particles, can be
separated into two equations, one describing the translational
motion of the molecule and the other its internal motion. The
latter has the form
+
r 2 dr\ drj^r 2 sin <3^ Sm d$) ^ r 2 sin 2 d<p 2
^r\W tf(r)}*«0, (353)
in which /x, the reduced mass, is given by the equation
M,M 2
anc
X36a] VIBRATION OF DIATOMIC MOLECULES 265
and r, #, <p are polar coordinates of the second nucleus relative to
the first as origin. In Section 18a it was also shown that this
equation can be separated into three equations in the three
variables <p } #, and r, respectively. The solution of the <p and #
equations, which are obtained in Sections 186, 18c, and 19, are
*m(<p) = — ^ iA " (355)
V27T
in which P^cos #) is an associated Legendre function (Sec. 196),
<£> and are the <p and # factors, respectively, in the product
function
iKr, #, <p) = #(r)e(t>)*(*). (357)
Instead of the azimuthal quantum number I, used for the hydro
gen atom, we have here adopted the letter K, and for the magnetic
quantum number m we here use M, in agreement with the usual
notation for molecular spectra. Both M and K must be integers,
for the reasons discussed in Sections 186 and 18c, and, as there
shown, their allowed values are
K = 0, 1, 2, • • • ; M = K, K + 1, • • , K  1, K.
(358)
Just as in the case of hydrogen, the quantum numbers M and K
represent angular momenta (see also Sec. 52), the square of the
total angular momentum due to the rotation of the molecule 1
being
K(K + 1)£,, (359)
while the component of this angular momentum in any specially
chosen direction (taken as the z direction) is
M~ (3510)
In Section 40d it will be shown that dipole radiation is emitted
or absorbed only for transitions in which the quantum number
1 There may be additional angular momentum due to the electrons.
266 THE ROTATION AND VIBRATION OF MOLECULES [X36b
K changes by one unit; i.e., the selection rule for K is
AK = ±1.
Likewise, the selection rule for M is
AM = or ±1.
The energy of the molecule does not depend on M (unless there
is a magnetic field present), so that this rule is not ordinarily
of importance in the interpretation of molecular spectra.
The equation for R(r) (Eq. 1826) is
^•t>+[ KJ ^ ±)+ > u ™
R = 0,
(3511)
in which for simplicity wc have omitted the subscripts n and v.
This may be simplified by the substitution
R(r) = is(r), (3512)
which leads to the equation
d*S
dr*
+ [ I ^±^+ S ^iWU(r)}]s = 0. (3513)
35b. The Nature of the Electronic Energy Function. — The
solution of the radial equation 3513 involves a knowledge of
the electronic energy function U(r) discussed in Section 34. The
theoretical calculation of U(r) requires the solution of the wave
equation for the motion of the electrons, a formidable problem
which has been satisfactorily treated only for the very simplest
molecules, such as the hydrogen molecule (Sec. 43). It is
therefore customary to determine U(r) empirically by assuming
some reasonable form for it involving adjustable parameters
which are determined by a comparison of the observed and calcu
lated energy levels.
From the calculations on such simple molecules as the hydrogen
molecule and from the experimental results, we know that
U(r) for a stable diatomic molecule is similar to the function
plotted in Figure 342. When the atoms are very far apart
(r large), the energy is just the sum of the energies of the two
individual atoms. As the atoms approach one another there
X35c] VIBRATION OF DIATOMIC MOLECULES 267
is for stable states a slight attraction which increases with
decreasing r, as is shown by the curvature of U in Figure 342.
For stable molecules, U must have a minimum value at the
equilibrium separation r = r e . For smaller values of r, U rises
rapidly, corresponding to the high repulsion of atoms "in contact/ 7
For most molecules in their lower states of vibration it will be
found that the wave function has an appreciable value only in a
rather narrow region near the equilibrium position, this having
the significance that the amplitude of vibration of most molecules
is small compared to the equilibrium separation. This is impor
tant because it means that for these lower levels the nature of
the potential function near the minimum is more important than
its behavior in other regions.
However, for higher vibrational levels, that is, for larger ampli
tudes of vibration, the complete potential function is of impor
tance. The behavior of U in approaching a constant value for
larger values of r is of particular significance for these higher
levels and is responsible for the fact that if sufficient energy is
transferred to the molecule it will dissociate into two atoms.
In the following sections two approximations for U(r) will be
introduced, the first of which is very simple and the second
somewhat more complicated but also more accurate.
35c. A Simple Potential Function for Diatomic Molecules. —
The simplest assumption which can be made concerning the force
between the atoms of a diatomic molecule is that it is proportional
to the displacement of the internuclear distance from its equilib
rium value r e . This corresponds to the potential function
U(r) = V 2 k(r  r e )\ (3514)
which is plotted in Figure 351. k is the force constant for the
molecule, the value of which can be determined empirically from
the observed energy levels. A potentialenergy function of this
type is called a Hooke'slaw potential energy function.
It is obvious from a comparison of Figures 34r2 and 351
that this simple function is not at all correct for large internuclear
distances. Nevertheless, by a proper choice of k a fair approxi
mation to the true U{r) can be achieved in the neighborhood
of r = r e . This approximation corresponds to expanding the
true U(r) in a Taylor series in powers of (r — r e ) and neglecting
all powers above the second, a procedure which is justified only
268 THE ROTATION AND VIBRATION OF MOLECULES [X35c
for small values of r — r . The coefficient of (r — r e )° (that is,
the constant term) in this expansion can be conveniently set
equal to zero without loss of generality so far as the solution
of the wave equation is concerned. The linear term in the
u(r)
Fig. 361. — Hooke'slaw potential function as an approximation to U (r).
expansion vanishes, inasmuch as U(r) has a minimum at r = r 6 ,
\(d 2 Jj\
and so the series begins with the term o\~J~2j ( r ~" r «) 2 
Comparison with Equation 3514 shows that the force constant
k is equal to ( j^ )
Insertion of this form for U(r) into the radial equation 3513
yields the equation
d*S
dr*
+
[
K( K + 1) , 8tt
~2 I
^
rfc(r  r e )
■}>  °>
(3515)
which may be transformed by the introduction of the new inde
pendent variable p = r — r e (the displacement from the equilib
rium separation) into the equation
** + *^iw  **„«  » K(K ± l) \s =
Since the approximation which we have used for U(r) is good only
for p small compared to r«, it is legitimate to introduce the
expansion
X88c] VIBRATION OF DIATOMIC MOLECULES 269
a step which leads to the result
P + j^{ w  K ( K + IV + 2K ( K + Djtp 
3K(K + \)~y  kp^S = 0, (3516)
in which powers of p/r, greater than the second have been
neglected, and the symbol a has been introduced, with
and I e — v>r*. I e is called the equilibrium moment of inertia of
the molecule.
By making a suitable transformation p — $ \ a, we can
eliminate the term containing the first power in the independent
variable, obtaining thereby an equation of the same form as
Equation 111, the wave equation for the harmonic oscillator,
which we have previously solved. It is easily verified that the
proper value for a is
SK(K + 1> + V 2 krf
and that the introduction of this transformation into Equation
3516 yields the equation
d*S S^(\ {K(K + !)„]* 1
 [jfc + 3K(K + l)^]f 2 }S = 0. (3518)
We seek the solutions of this equation which make ^(r, #, <p) of
Equation 357 a satisfactory wave function. This requires
that S vanish for r = and r = oo , the former condition entering
because of the relation R = S. We know the solutions of the
r
equation which vanish f or f = — oo and f = + <*> , since for these
boundary conditions the problem is analogous to that of the
linear harmonic oscillator (Sec. 11). Because of the rapid
270 THE ROTATION AND VIBRATION OF MOLECULES [X35c
decrease in the harmonic oscillator functions outside of the
classically permitted region (see Fig. 113), it does not introduce
a serious error to consider that the two sets of boundary conditions
are practically equivalent, so that as an approximation we may
use the harmonic oscillator wave functions for the functions S.
The energy levels are, therefore, using the results of Section
11a,
W„ K  K(K + 1),  V<f+^, + („ + l)w,
(3519)
in which
1 [ krl + QKjK + l)<r\ H
2tt\ /ir»
r
(3520)
and v is the vibrational quantum number (corresponding to the
quantum number n for the harmonic oscillator), which can take
on the values 0, 1, 2, • • • . The functions £(f) are (Sec. 11)
Sr(f) = {(f)^}^ _fr! ^ (V " r) ' (35_21)
in which a — 4w 2 nv' e /h and f = p — a = r — r e — a, and H v is
the tfth Hermite polynomial.
The values of fc, r e , and <r for actual molecules are of such
magnitudes that the expression for W can be considerably simpli
fied without loss of accuracy by the use of the expansions
1 = 1 L _ SK(K + l )cr \
3K(K + l)<r + l Akr 2 } 2 kr 2 \ ' kr 2 e ~~ "^ /'
l/^r e 2 +6K(^ + lV\ ,/2
"° 27r\ ~^1 J
Introducing these into Equation 3519, we obtain for W the
expression
{K(K + l)v\*
(v + 0*,.
W 9 .k = U + £ ]A„. + #(# + l)«r
X2*
in which only the first terms of the expansions have been used
and the symbol v 6 is given by
X35d] VIBRATION OF DIATOMIC MOLECULES 271
1 Ik
(3522)
Replacing k by its expression in terms of v, and introducing
the value of Equation 3517 for a, we finally obtain for W the
expression
IP..* = ^ + a j*'. + W + Dg^  i28^7r " (35 " 23)
The first term is evidently the vibrational energy of the mole
cule, considered as a harmonic oscillator. The second term is
the energy of rotation, assuming that the molecule is a rigid
body, 1 while the third term is the correction which takes account
of the stretching of the actual, nonrigid molecule due to the
rotation. The terms of higher order are unreliable because
of the inaccuracy of the assumed potential function.
The experimental data for most molecules fit Equation 3523
fairly well. For more refined work additional correction terms
are needed, one of which will be obtained in the next section.
35d. A More Accurate Treatment. The Morse Function. —
The simple treatment which we have just given fails to agree
with experiment in that it yields equally spaced levels, whereas
the observed vibrational levels show a convergence for increasing
values of v. In order to obtain this feature a potential function
U(r) is required which is closer to the true U(r) described in
1 This is seen by allowing k to become infinite, causing the third term to
vanish (because y e — ►«>). A rigid molecule would have no vibrational
energy, so the first term would become an additive constant. The rigid
rotator is often discussed as a separate problem, with the wave equation
1 d ( . dA 1 dV 8tt 2 /
( sin t?— + ~ + — WV =0, (3524)
sin#ch>\ <W/ sin 2 tfd<? 2 h 2
the solutions of which are \j/ = &m (<p)Qkm(&), in which $ and are given by
h 2
Equations 355 and 356. The energy levels are Wk = K(K + 1) •
Sir 2 !
The rigid rotator is of course an idealization which does not occur in nature.
Another idealized problem is the rigid rotator in a plane, for which the
wave equation is
£ + » w *  ° (35  25)
The solutions are $ = sin M*p and ^ = cos M<p, M = 0, 1, 2, • • • , and
the energy levels are Wm = M 2 h 2 /Sw 2 T (Sec. 25a).
272 THE ROTATION AND VIBRATION OF MOLECULES [X35d
Section 356, especially with regard to its behavior for large
values of r.
Morse 1 proposed a function of the form
U(r) = D{1  e^^} 2 , (35^26)
which is plotted in Figure 34r2. It has a minimum value of
zero at r = r e and approaches a finite value D for r large. It
therefore agrees with the qualitative considerations of Section 356
except for its behavior at r = 0. At this point the true U(r)
is infinite, whereas the Morse function is finite. However, the
Morse function is very large at this point, and this deficiency
is not a serious one.
With the introduction of this function, the radial equation
3513 becomes
« + {«*£« + »*ori>i»r~~ +
2De
Ur~r )
If we make the substitutions
y = e airr.) an( J
\s = 0.
A = K{K + 1)
8xV. 2
the radial equation becomes
a 2 n 2 \ jr y j/ 2 ry
dy* y dy
(3527)
(3528)
(3529)
The quantity r%/r 2 may be expanded in terms of y in the following
way: 2
05?)'
= i + ~(v
ar e
1 >+(k + ^ 1)f
+
(3530)
the series being the Taylor expansion of the second expression
in powers of (y — 1). Using the first three terms of this expan
sion in Equation 3529 we obtain the result
1 P. M. Morse, Phys. Rev. 34, 57 (1929).
1 This treatment is due to C. L. Pekeris, Phys. Rev. 46, 98 (1934).
solved the equation for the case K = only.
Morse
X36d]
d*S
VIBRATION OF DIATOMIC MOLECULES
273
IdS + **(W^Dc. + 2Dc 1 _ D _ \
ydy a 2 h 2 \ y 2 V /
in which
C2 = A(i+jy.
(3531)
(3532)
The substitutions
S(y) = e~W(z),
2 = 2dy,
8
TC'H
«P=^H7P+«0,
6 2 = 
a 2 /i 2
32*r
(3533)
a 2 A
^(TTZ)Co)j
simplify Equation 3531 considerably, yielding the equation
dW
dz
.t +
in which
C^  >)
4**n
a 2 h 2 d
f + 'FO,
dz z
1,
{2D  Cl )  5(6 + 1)
(3534)
(3535)
Equation 3534 is closely related to the radial equation 1837
of the hydrogen atom and may be solved in exactly the same
manner. If this is done, it is found that it is necessary to
restrict v to the values 0, 1, 2, • • • in order to obtain a poly
nomial solution. ' If we solve for W by means of Equations 3535
and the definitions of Equations 3533, 3532, and 3528, we
obtain the equation
Wk., = D+c 
(D + a)
2 ah(D  V 2Cl )
7rVWl> + c,\ V
oW/ , IV
1 The solutions for v integral satisfy the boundary conditions F — ► as
r _► _ oo instead of as r — ► (Sec. 36c).
274 THE ROTATION AND VIBRATION OF MOLECULES [X36c
By expanding in terms of powers of ci/D and c 2 /Z>, this relatior
may be brought into the form usually employed in the study ol
observed spectra; namely,
TT = *(" + 5)  x **l? + tj + K(K + 1)Be +
D e K\K + l) 2  a e (v + V 2 )K(K + 1), (3536)
in which c is the velocity of light, and 1
a J2D
Xe ~~£D'
B e = g^, ) (3537)
a e =
167rV?Z>'
ir*D\ar e a 2 r 2 J
For nearly all molecules this relation gives very accurate
values for the energy levels; for a few molecules only is it neces
sary to consider further refinements.
We shall not discuss the wave functions for this problem.
They are given in the two references quoted.
Problem 351. Another approximate potential function which has beer
used for diatomic molecules 2 is
B Ze*
U(r) « 
r 2 r
Obtain the energy levels for a diatomic molecule with such a potential func
tion, using the polynomial method. (Hint: Follow the procedure of Sec. IS
closely.) Expand the expression for the energy so obtained in powers o)
07T 2 /i
(K + l) 2 ~r~^ an d compare with Equation 3523. Also obtain the positioi
h 2 B
of the minimum of U(r) and the curvature of U(r) at the minimum.
Problem 362. Solve Equation 3535 for the energy levels.
1 The symbol o>« is often used in place of £«.
2 E. Ftjes, Ann. d. Phys. 80, 367 (1926).
X86a] THE ROTATION OF POLYATOMIC MOLECULES 275
36. THE ROTATION OF POLYATOMIC MOLECULES
The straightforward way to treat the rotational and vibrational
motion of a polyatomic molecule would be to set up the wave
equation for ^ ntV (£) (Eq. 344), introducing for U n (i) an expres
sion obtained either by solution of the electronic wave equation
343 or by some empirical method, and then to solve this nuclear
wave equation, using some approximation method if necessary.
This treatment, however, has proved to be so difficult that it is
customary to begin by making the approximation of neglecting
all interaction between the rotational motion and the vibrational
motion of the molecule. 1 The nuclear wave equation can then
be separated into two equations, one, called the rotational wave
equation, representing the rotational motion of a rigid body.
In the following paragraphs we shall discuss this equation,
first for the special case of the socalled symmetricaltop molecules,
for which two of the principal moments of inertia are equal
(Sec. 36a), and then for the unsymmetricaltop molecules, for
which the three principal moments of inertia are unequal (Sec.
366). The second of the two equations into which the nuclear
wave equation is separated is the vibrational wave equation,
representing the vibrational motion of the nonrotating molecule.
This equation will be treated in Section 37, with the usual
simplifying assumption of Hooke'slaw forces, the potential
energy being expressed as a quadratic function of the nuclear
coordinates.
36a. The Rotation of Symmetricaltop Molecules. — A rigid
body in which two of the three principal moments of inertia 2
iSee, however, C. Eckart, Phys. Rev. 47, 552 (1935); J. H. Van Vleck,
ibid. 47, 487 (1935); D. M. Dennison and M. Johnson, ibid. 47, 93 (1935).
2 Every body has three axes the use of which permits the kinetic energy
to be expressed in a particularly simple form. These are called the principal
axes of inertia. The moment of inertia about a principal axis is denned by
the expression Jpr 2 dr, in which p is the density of matter in a given volume
element dr, r is the perpendicular distance of this element from the axis iii
question, and the integration is over the entire volume of the solid. For a.
discussion of this question see J. C. Slater and N. II. Frank, " Introduction
to Theoretical Physics," p. 94, McGrawHill Book Company, Inc., New
York, 1933.
In case that a molecule possesses an nfold symmetry axis with n greater
than 2 (such as ammonia, with a threefold axis), then two principal moments
of inertia about axes perpendicular to this symmetry axis are equal, and the
276 THE ROTATION AND VIBRATION OF MOLECULES [X36a
are equal is called a symmetrical top. Its position in space is
best described by the use of the three Eulerian angles #, «*>, and x
shown in Figure 361. # and v are the ordinary polarcoordinate
angles of the axis of the top while x (usually called \p) is the
angle measuring the rotation about this axis.
Since we have considered only assemblages of point particles
heretofore, we have not given the rules for setting up the wave
equation for a rigid body. We shall not discuss these rules
here 1 but shall take the wave equation for the symmetrical top
Fia. 361. — Diagram showing Eulerian angles.
from the work of others. 2 Using C to represent the moment of
inertia about the symmetry axis and A the two other equal
moments of inertia, this wave equation is
molecule is a symmetrical top. A twofold axis does not produce a symmet
ricaltop molecule (example, water). If the molecule possesses two or more
symmetry axes with n greater than 2, it is called a sphericaltop molecule,
all three moments of inertia being equal.
1 Since the dynamics of rigid bodies is based on the dynamics of particles,
these rules must be related to the rules given in Chapter IV. For a dis
cussion of a method of finding the wave equation for a system whose Hamil
tonian is not expressed in Cartesian coordinates, see B. Podolsky, Phys.
Rev. 32, 812 (1928), and for the specific application to the symmetrical
top see the references below.
2 F. Reiche and H. Rademacher, Z. f. Phys. 39, 444 (1926); 41, 453
(1927); R. de L. Kroniq and I. I. Rabi, Phys. Rev. 29, 262 (1927). D. M.
Dennison, Phys. Rev. 28, 318 (1926), was the first to obtain the energy
levels for this system, using matrix mechanics rather than wave mechanics.
X36a) THE ROTATION OF POLYATOMIC MOLECULES 277
+
K*"8)
1
sin 2
/cos'
\sin
2 #
2 #
<

2cos#
sin 2 #
ay
+
Br* AW,
sin#d#\
(361)
The angles x and <p do not occur in this equation, although
derivatives with respect to them do. They are therefore cyclic
coordinates (Sec. 17), and we know that they enter the wave
function in the following manner:
^ = e(d)e iM *e iK x, (362)
in which M and K have the integral values 0, ±1, ±2, • * • .
Substitution of this expression in the wave equation confirms
this, yielding as the equation in #
1 d ( . d&\ j_M^ , /cos 2 * JA 2
Sr7# ^V Sm 3*/ ~ W 2 * + Un 2 t> + Cr
 2^W  *£w)e = 0. (3(W)
sm 2 # h 2 )
We see that # = and # = 7r are singular points for this equation
(Sec. 17). It is convenient to eliminate the trigonometric
functions by the change of variables
x = y 2 {\  cos #U
e(*) = r(x), I
at the same time introducing the abbreviation
(364)
x = S^W _ A^ 2> (36 _ 5)
the result being
U*  *>£} + i x  "t$*:> 1), >  ° <»
The singular points, which are regular points, have now been
shifted to the points and 1 of x, so that the indicial equation
must be obtained at each of these points. Making the sub
stitution T(x) = x a G(x), we find by the procedure of Section 17
that s equals Vi\K — M\, while the substitution
T(x) = (1  xYH{\  x)
278 THE ROTATION AND VIBRATION OF MOLECULES [X36a
yields a value of }4\K + M\ for s'. Following the method of
Section 18c we therefore make the substitution
0(0) = T(x) = xW K ~ Ml (l  *)W*+*iF(a;), (367)
which leads to the equation 1 for F
x(l  x)g + (a  0x)^ + 7^=0, (368)
in which
a = \K  M\ + 1,
P = \K + M\ + \K M\+2.
and
7 = X + JC*  (Ml* + m\+ y 2 \K  m\kv 2 \k + M\ +
y 2 \K m\ + i).
We can now apply the polynomial method to this equation by
substituting the series expression
F(x) = ^a„z"
in Equation 368. The recursion formula which results is
For this to break off after the jth term (the series is not an
acceptable wave function unless it terminates), it is necessary
for the numerator of Equation 369 to vanish, a condition which
leads to the equation for the energy levels
in which
j = j + y 2 \K + M\ + y 2 \K  M\ t (3611)
that is, J is equal to, or larger than, the larger of the two quan
tities \K\ and \M\. The quantum number J is therefore zero or a
positive integer, so that we have as the allowed values of the
three quantum numbers
J = 0, 1, 2, • • • , )
K = 0, ±1, ±2, • • • , ±J\ (3612)
M =0, ±1, ±2, • • • , ±J.)
1 This equation is well known to mathematicians as the hypergeomelric
equation.
XS6a] THE ROTATION OF POLYATOMIC MOLECULES
279
h 2
It can be shown that J(J + l)l~2 k *ke s Q^are of the total
angular momentum, while Kh/2ir is the component of angular
momentum along the symmetry axis of the top and Mh/2v the
component along an arbitrary axis fixed in space.
—55
Wt
•54
— 55
53
•44
•52
51
43
50
4E
33,41
32
21
•22
21
30
20
A*2C
00
•54
53
52 r
S.50
44
•43
—42
41
40
33
32
,31
30
22
Ji10
AiC
00
Fio. 3&2. — Energylevel diagram for symmetricaltop molecule, with A = 2C
and with ii » J^C. Values of the quantum numbers / and K are given for
each level.
When K is zero, the expression for W reduces to that for the
simple rotator in space, given in a footnote in Section 35c. The
energy does not depend on M or on the sign of K, and hence
the degeneracy of a level with given J is 2 J + 1 or 4 J + 2, depend
ing on whether K is equal to zero or not. The appearance 1 of
the set of energy levels depends on the relative magnitudes of
A and C, as shown in Figure 3&2.
1 For a discussion of the nature of these energy levels and of the spectral
lines arising from them, see D. M. Dennison, Rev. Mod. Phy$. 3, 280 (1931).
280 THE ROTATION AND VIBRATION OF MOLECULES [X8«b
The wave functions can be constructed by the use of the
recursion formula 369. In terms of the hypergeometric
functions 1 F(a, b; c;x), the wave function is
+jkm(#, <p, x) = N JKU x^ K  m {l  a.)W*+*l e «*H*x>
F(J + MP  1,J + MP;1 + \K  M\; x), (3613)
in which x = \i{\ — cos d) and
v _ ( _ ( 2J+l)(J+y 2 \K+M\+V 2 \KM\)\
^">>\Sw*(Jyi\K+M\H\KM\)\(\KM\\)*
(Jy^K+Ml+mKMWX*
(J + y 2 \K+M\y 2 \KM\)\) ^° iV
In case that all three principal moments of inertia of a molecule
are equal, the molecule is called a sphericaltop molecule (examples:
methane, carbon tetrachloride, sulfur hexafluoride). The energy
levels in this case assume a particularly simple form (Problem
362).
It has been found possible to discuss the rotational motion of
molecules containing parts capable of free rotation relative to
other parts of the molecule. Nielsen 2 ha? treated the ethane
molecule, assuming the two methyl groups to rotate freely
relative to one another about the CC axis, and La Coste 3 has
similarly discussed the tetramethylmethane molecule, assuming
free rotation of each of the four methyl groups about the axis
connecting it with the central carbon atom.
Problem 361. Using Equation 369, construct the polynomial F(x)
for the first few sets of quantum numbers.
Problem 362. Set up the expression for the rotational energy levels for
a sphericaltop molecule, and discuss the degeneracy of the levels. Calcu
late the term values for the six lowest levels for the methane molecule,
assuming the CH distance to be 1 .06 A.
36b. The Rotation of Unsymmetricaltop Molecules. — The
treatment of the rotational motion of a molecule with all three
principal moments of inertia different (called an unsymmetrical
top molecule) is a much more difficult problem than that of the
preceding section for the symmetrical top. We shall outline a
1 The hypergeometric function is discussed in Whittaker and Watson,
"Modern Analysis," Chap. XIV.
" H. H. Nielsen, Phys. Rev. 40, 445 (1932).
*L. J. B. La Coste. Phys. Rev. 46. 718 (1934).
X36b] THE ROTATION OF POLYATOMIC MOLECULES 281
procedure which has been used with success in the interpretation
of the spectra of molecules of this type.
Let us write the wave equation symbolically as
Hyp = W$. (3615)
Inasmuch as the known solutions of the wave equation for a
symmetricaltop molecule form a complete set of orthogonal
functions (discussed in the preceding section), we can expand
the wave function \p in terms of them, writing
* = % a JKM r JKM , (3616)
JKM
in which we use the symbol \p] KM to represent the symmetricaltop
wave functions for a hypothetical molecule with moments of
inertia A 0y B (= A ), and C . If we now set up the secular
equation corresponding to the use of the series of Equation 3616
as a solution of the unsymmetricaltop wave equation (Sec. 27a),
we find that the only integrals which are not zero are those
between functions with the same values of J and M y so that
the secular equation immediately factors into equations corre
sponding to variation functions of the type 1
4 J
ypjoM = 2) a JK*tfjKM' (3617)
On substituting this expression in the wave equation 3615,
we obtain the equation
%a K H+° K = WX*** * ( 36 ~ 18 )
K K
in which for simplicity we have omitted the subscripts J and M,
the argument from now on being understood to refer to definite
values of these two quantum numbers. On multiplication by
yf/\* and integration, this equation leads to the following set of
simultaneous homogeneous linear equations in the coefficients a K :
s
a K (H L KtLKW)=0, L=J,J+l, •••,+/, (3619)
in which 8lk has the value 1 f or L = K and otherwise, and H L k
1 The same result follows from the observation that J and M correspond
to the total angular momentum of the system and its component along a
fixed axis in space (see Sec. 52, Chap. XV).
282 THE ROTATION AND VIBRATION OF MOLECULES [X87a
represents the integral JVi *£ty£dr. This set of equations has a
solution only for values of W satisfying the secular equation
Hj,j — W Hj,j+i
H Jt j
Hj+i,j
Hj,j  W
= 0.
(3620)
These values of W are then the allowed values for the rotational
energy of the unsymmetricaltop molecule. Wang 1 has evaluated
the integrals Hlk and shown that the secular equation can
be further simplified. The application in the interpretation
of the rotational fine structure of spectra has been carried
out in several cases, including water, 2 hydrogen sulfide, 3 and
formaldehyde. 4
37. THE VIBRATION OF POLYATOMIC MOLECULES
The vibrational motion of polyatomic molecules is usually
treated with an accuracy equivalent to that of the simple dis
cussion of diatomic molecules given in Section 35c, that is, with
the assumption of Hooke'slaw forces between the atoms. When
greater accuracy is needed, perturbation methods are employed.
Having made the assumption of Hooke'slaw forces, we employ
the method of normal coordinates to reduce the problem to soluble
form. This method is applicable whether we use classical
mechanics or quantum mechanics. Inasmuch as the former
provides a simpler introduction to the method, we shall consider
it first.
37a. Normal Coordinates in Classical Mechanics. — Let the
positions of the n nuclei in the molecule be described by giving
the Cartesian coordinates of each nucleus referred to the
equilibrium position of that nucleus as origin, as shown in Figure
371. Let us call these coordinates q[ } q' 2r • • • , q' Zn . In terms
of them we may write the kinetic energy of the molecule in the
form
1 S. C. Wang, Phys. Rev. 34, 243 (1929). See also H. A. Kramers and
G. P. Ittmann, Z. f. Phys. 53, 553 (1929); 68, 217 (1929); 60, 663 (1930);
O. Klein, Z. f. Phys. 58, 730 (1929); E. E. Witmer, Proc. Not. Acad. Sci.
13, 60 (1927); H. H. Nielsen, Phys. Rev. 38, 1432 (1931).
> R. Mecke, Z. f. Phys. 81, 313 (1933).
8 P. C. Cross, Phys. Rev. 47, 7 (1935).
4 G. H. Dieke and G. B. Kistiakowskt, Phys. Rev. 46, 4 (1934).
XS7a] THE VIBRATION OF POLYATOMIC MOLECULES
3»
TH%M&;
283
(371)
t1
in which M »• is the mass of the nucleus with coordinate q[. By
changing the scale of the coordinates by means of the relation
qi = y/Mrfi, i = 1, 2, • • • , 3n, (372)
we can eliminate the masses from the kinetic energy expression,
obtaining
T = yi%& (373)
The potential energy V depends on the mutual positions of
the nuclei and therefore upon the coordinates g». If we restrict
Origin
*2
6
Origin
9
Origin
# *7 ^6 *9
— >8
Fia. 371. — Coordinates q[ . . . q** of atoms measured relative to equilibrium
positions.
ourselves to a discussion of small vibrations, we may expand
V as a Taylor series in powers of the g's,
Vfatf*
<7s») = V + 2^j «< + gS 6 '**' + " " ' '
(874)
284 THE ROTATION AND VIBRATION OF MOLECULES [X37a
in which &,,• is given by
ba
bii ~ \dlidqJo'
and the subscript means that the derivatives are evaluated
at the point q\ = 0, g 2 = 0, etc. If we choose our zero of energy
so that V equals zero when q h q 2j etc. are zero, then V is zero.
Likewise the second term is zero, because by our choice of coordi
nate axes the equilibrium position is the configuration qi = 0,
q 2 = 0, etc., and the condition for equilibrium is
(sa*
i = 1, 2, • • • , 3n. (375)
Neglecting higher terms, we therefore write
Viqiq* ' ' ' <Z3„) = MtyM,. (376)
v
Using the coordinates q^ we now set up the classical equations
of motion in the Lagrangian form (Sec. lc). In this case the
kinetic energy T is a function of the velocities qi only, and the
potential energy V is a function of the coordinates qi only, and
in consequence the Lagrangian equations have the form
On introducing the above expressions for T and V we obtain the
equations of motion
9k + %b ikqi = 0, k = 1, 2, • • • , 3n. (378)
In case that the potentialenergy function involves only squares
q% and no crossproducts q t gj with i 5* j; that is, if ba vanishes
for i 9^ j, then these equations of motion can be solved at once.
They have the form
qu + b kk q k =0, k = 1, 2, • • • , 3n, (379)
the solutions of which are (Sec. la)
9* = q$ sin (\/Sirf + «*)i * = If 2, • • • , 3n, (3710)
in which the gj's are amplitude constants and the 8 k s phase
constants of integration. In this special case, then, each of the
X37aJ THE VIBRATION OF POLYATOMIC MOLECULES 285
coordinates q k undergoes harmonic oscillation, the frequency
being determined by the constant b kk .
Now it is always possible by a simple transformation of
variables to change the equations of motion from the form
378 to the form 379; that is, to eliminate the crossproducts
from the potential energy and at the same time retain the form
373 for the kinetic energy. Let us call these new coordinates
Qi(l = 1, 2, • • • , 3n). In terms of them the kinetic and the
potential energy would be written
T = y 2 %Q? (3711)
i
and
V = V 2 %\iQl (3712)
i
and the solutions of the equations of motion would be
Qi = QJsin (VXit + 8i), I = 1, 2, • • • , 3n. (3713)
Instead of finding the equations of transformation from the
q's to the Q's by the consideration of the kinetic and potential
energy functions, we shall make use of the equations of motion.
In case that all of the amplitude constants Q$ are zero except one,
Qi; sa y> then Qi will vary with the time in accordance with
Equation 3713, and, inasmuch as the q's are related to the
Q's by the linear relation
3n
qk = X BklQl > (37 " 14)
each of the q's will vary with the time in the same way, namely,
q k = A k sin (V\t + 5 X ), k = 1, 2, • • • , 3n. (3715)
In these equations A k represents the product B k iQ[, and X the
quantity Xi, inasmuch as we selected Q\ as the excited coordinate;
the new symbols are introduced for generality. On substituting
these expressions in the equations of motion 378, we obtain the
set of equations
3n
\A k + %b ik Ai = 0, k = 1,2, • • • , 3n. (3716)
This is a set of 3n simultaneous linear homogeneous equations
in the 3n unknown quantities A k . As we know well by this time
286 THE ROTATION AND VIBRATION OF MOLECULES [X37a
(after Sees. 24, 26d, etc.), this set of equations possesses a solution
other than the trivial one A x = A 2 = # * • =0 only when the
corresponding determinantal equation (the secular equation of
perturbation and variation problems) is satisfied. This equa
tion is
6n — X 612 • • • biBn
b *\ &22 7 x ; ; ; bi * = o. (3717)
bznl Ozn2 ' ' * #8n8n ~~ X
In other words, Equation 3715 can represent a solution of the
equations of motion only when X has one of the 3n values which
satisfy Equation 3717. (Some of these roots may be equal.)
Having found one of these roots, we can substitute it in Equation
3716 and solve for the ratios 1 of the A's. If we put
A k i = B kl Ql (3718)
and introduce the extra condition
X B it = l > (3719)
k
in which the subscript I specifies which root \i of the secular
equation has been used, then we can determine the values of
the Bus, Q\ being left arbitrary.
By this procedure we have obtained 3n particular solutions
of the equations of motion, one for each root of the secular
equation. A general solution may be obtained by adding all
of these together, a process which yields the equations
3n
q k = X Q ° Bkl sin (v ^' + 5z)  (37 ~ 20)
This solution of the equations of motion contains 6n arbitrary
constants, the amplitudes Q° t and the phases 6i f which in
any particular case are determined from a knowledge of the
initial positions and velocities of the n nuclei.
We have thus solved the classical problem of determining the
positions of the nuclei as a function of the time, given any set
of initial conditions. Let us now discuss the nature of the
1 These equations are homogeneous, so that only the ratios of the A's can
be determined. The extra condition 3719 on the Bus then allows them
to be completely determined.
X37a] THE VIBRATION OF POLYATOMIC MOLECULES 287
solution. As mentioned above, if we start the molecule vibrating
in such a way that all the QJ's except one, say Q° u are zero, the
solution is
q k = Q\B kl sin (y/\ x t + h x ), k = 1, 2, • • • , 3n, (3721)
which shows that each of the nuclei carries out a simple harmonic
oscillation about its equilibrium position with the frequency
v x = ^. (3722)
All of the nuclei move with the same frequency and the same
phase; that is, they all pass through their equilibrium positions
at the same time and reach
their positions of maximum
amplitude at the same time.
These amplitudes, however,
are not the same for the differ
ent nuclei but depend on the
values of the B k i's and upon the
initial amplitude, which is
determined by Q?. A vibration
governed by Equation 3721
and therefore having these prop
erties is called a normal mode Fig. 372.™One of the normal modes
of vibration of a symmetrical triatomic
of Vibration of the System (see molecule. Each of the atoms moves in
Fiff 37—2^ an( * out a ^ ong a r& dial direction as shown
®* '" .ji by the arrows. All the atoms move with
It is not required, however, the same frequency and phase, and in this
that the nuclei have initial special case with the same amplitude.
amplitudes and velocities such that the molecule undergo such a
special motion. We can start the molecule off in any desired man
ner, with the general result that many of the constants Q\ will be
different from zero. In such a case the subsequent motion of
the nuclei may be thought of as corresponding to a superposition
of normal vibrations, each with its own frequency V%/27r and
amplitude Q?. The actual motion may be very complicated,
although the normal modes of vibration themselves are fre
quently quite simple.
The normal coordinates of the system are the coordinates Q h
which we introduced in Equation 3714. These coordinates
specify the configuration of the system just as definitely as the
original coordinates q%.
288 THE ROTATION AND VIBRATION OF MOLECULES [X37b
The expansion of V given in Equation 374 is not valid except
when the nuclei stay near their equilibrium positions. That is,
we have assumed that the molecule is not undergoing transla
tional or rotational motion as a whole. Closely related to this
is the fact, which we shall not prove, that zero occurs six 1 times
among the roots X* of the secular equation. The six normal
modes of motion corresponding to these roots, which are not
modes of vibration because they have zero frequency, are the
three motions of translation in the x ) y, and z directions and
the three motions of rotation about the x } y ) and z axes.
37b. Normal Coordinates in Quantum Mechanics. — It can be
shown 2 that when the coefficients B k i of Equation 3714 are
determined in the manner described in the last section, the
introduction of the transformation 3714 for the g^'s into the
expression for the potential energy yields the result
V = KX b »M> = ^2 XlQ ?; (3723)
that is, the transformation to normal coordinates has eliminated
the crossproducts from the expression for the potential energy.
In addition, this transformation has the property of leaving the
expression for the kinetic energy unchanged in form; 3 i.e.,
T = «2«  «SQf. (37 _ 24 )
These properties of the normal coordinates enable us to treat the
problem of the vibrations of polyatomic molecules by the
methods of quantum mechanics.
The wave equation for the nuclear motion of a molecule is
2i v ^ + *w {w ~ F) * = °> (37 ~ 25)
ii
in which ^ represents the nuclear wave function ^ n ,*(£) of Equa
1 This becomes five for linear molecules, which have only two degrees of
rotational freedom.
2 For a proof of this see E. T. Whittaker, "Analytical Dynamics,"
Sec. 77, Cambridge University Press, 1927.
3 A transformation which leaves a simple sum of squares unaltered is
called an orthogonal transformation.
X37bl THE VIBRATION OF POLYATOMIC MOLECULES 289
tion 344. In terms of the Cartesian coordinates g previously
described (Fig. 371), we write
n 3n
;  1 t  1
By changing the scale of the coordinates as indicated by Equation
372 we eliminate the APs, obtaining for the wave equation the
expression
3n
2fS+^VF)* = 0. (3727)
1 = 1
We now introduce the normal coordinates Qi. The reader can
easily convince himself that an orthogonal transformation will
leave the form of the first sum in the wave equation unaltered,
so that, using also Equation 3723, we obtain the wave equation
in the form
3n 3n
1=1 /=1
This equation, however, is immediately separable into 3n
onedimensional equations. We put
* = MQMQt) • • • lMQ«»), (3729)
and obtain the equations
m+^ w > ¥**)*> °> (37 " 30)
each of which is identical with the equation for the onedimen
sional harmonic oscillator (Sec. lla). The total energy W is
the sum of the energies Wk associated with each normal coordi
nate; that is,
3n
W = 2jTT*. (3731)
The energy levels of the harmonic oscillator were found in
Section lla to have the values (v + M)hv , where v is the quan
tum number and v the classical frequency of the oscillator.
Applying this to the problem of the polyatomic molecule, we
see that
290 THE ROTATION AND VIBRATION OF MOLECULES [X38
W = %W k = %(v k + V 2 )hv k} (37 _ 3 2)
jfc k
in which v k is the quantum number (v k = 0, 1, 2, • • • ) and v k \a
the classical frequency of the kth normal mode of vibration.
We have already seen (from Eq. 3722) that
v k = ^ (3733)
The energylevel diagram of a polyatomic molecule is therefore
quite complex. If, however, we consider only the fundamental
frequencies emitted or absorbed by such a molecule; that is, the
frequencies due to a change of only one quantum number v k
by one unit, we see that these frequencies are v u v 2 , • • • , vz n )
that is, they are the classical frequencies of motion of the
molecule.
This type of treatment has been very useful as a basis for the
interpretation of the vibrational spectra of polyatomic molecules.
Symmetry considerations have been widely employed to simplify
the solution of the secular equation and in that connection the
branch of mathematics known as group theory has been very
helpful. 1
38. THE ROTATION OF MOLECULES IN CRYSTALS
In the previous sections we have discussed the rotation and
vibration of free molecules, that is, of molecules in the gas phase.
There is strong evidence 2 that molecules and parts of molecules
in many crystals can rotate if the temperature is sufficiently
high. The application 23 of quantum mechanics to this problem
has led to a clarification of the nature of the motion of a molecule
within a crystal which is of some interest. The problem is
closely related to that dealing with the rotation of one part of a
molecule relative to the other parts, such as the rotation of
methyl groups in hydrocarbon molecules. 4
1 C. J. Brbster, Z. f. Phys. 24, 324 (1924); E. Wigner, Gottinger Nachr.
133 (1930); G. Placzek, Z. f. Phys. 70, 84 (1931); E. B. Wilson, Jr.,
Phys. Rev. 46, 706 (1934); /. Chem. Phys. 2, 432 (1934); and others.
*L. Pauling, Phys. Rev. 36, 430 (1930). This paper discusses the
mathematics of the plane rotator in a crystal as well as the empirical evidence
for rotation.
8 T. E. Stern, Proc. Roy. Soc. A 130, 551 (1931).
4 E. Teller and K. Weigert, Gdttinger Nachr. 218 (1933); J. E.
LennardJones and H. H. M. Pike, Trans. Faraday Soc. 30, 830 (1934).
X38] THE ROTATION OF MOLECULES IN CRYSTALS 291
The wave equation for a diatomic molecule in a crystal, con
sidered as a rigid rotator, obtained by introducing V into the
equation for the free rotator given in a footnote of Section 35c, is
1
sin#
£(3)
+
l av
+^v
V)+ = 0, (381)
sin 2 # dip 2
in which # and <p are the polar coordinates of the axis and J is
the moment of inertia of the molecule. The potential function V
is introduced as an approximate description of the effects of the
other molecules of the crystal upon the molecule in question.
2V 
Fio. 381. — Idealized potential function for a symmetrical diatomic molecule in
a crystal.
If the molecule being studied is made up of like atoms, such as
is O2 or H 2 , then a reasonable form to assume for V is
V = 7 (1  cos 20), (382)
which is shown in Figure 381. Turning a symmetrical molecule
end for end does not change 7, as is shown in the figure by the
periodicity of V with period w.
The wave equation 381 with the above form for V has been
studied by Stern, 1 who used the mathematical treatment given
by A. H. Wilson. 2 We shall not reproduce their work, although
the method of solution is of some interest. The first steps are
exactly the same as in the solution of the equation discussed in
Section 18c except that a threeterm recursion formula is obtained.
The method of obtaining the energy levels from this threeterm
formula is then similar to the one which is discussed in Section
42c, where a similar situation is encountered.
We have referred to the case of free rotation of methyl groups at the end of
Section 36a.
1 T. E. Stern, loc. cit.
1 A. H. Wilson, Proc. Roy. Soc. A 118, 628 (1928).
292 THE ROTATION AND VIBRATION OF MOLECULES [X38
The results obtained may best be described by starting with
the two limiting cases. When the energy of the molecule is
small compared with F (i.e., at low temperatures), then the
potential function can be regarded as parabolic in the neighbor
hood of the minima and we expect, as is actually found, that the
energy levels will be those of a twodimensional harmonic
oscillator and that the wave functions will show that the molecule
oscillates about either one of the two positions of equilibrium,
with little tendency to turn end over end. When the molecule
is in a state with energy large compared with V (i.e., at high
temperatures), the wave functions and energy levels approximate
those of the free rotator (Sec. 35c, footnote), the endoverend
motion being only slightly influenced by the potential energy.
In the intermediate region, the quantummechanical treatment
shows that there is a fairly sharp but nevertheless continuous
transition between oscillation and rotation. In other words, for
a given energy there is a definite probability of turning end over
end, in sharp contrast with the results of classical mechanics,
which are that the molecule either has enough energy to rotate
or only enough to oscillate.
The transition between rotation and oscillation takes place
roughly at the temperature T = 2Vo/kj where k is Boltzmann's
constant. This temperature lies below the melting point for a
number of crystals, such as hydrogen chloride, methane,
and the ammonium halides, and is recognizable experimentally
as a transition point in the heatcapacity curve. For solid
hydrogen even the lowest energy level is in the rotational region,
a fact which is" of considerable significance in the application of
the third law of thermodynamics.
Problem 381. Considering the above system as a perturbed rigid rota
tor, study the splitting of the rotator levels by the field, indicating by an
energylevel diagram the way in which the components of the rotator levels
begin to change as the perturbation is increased.
General References
A. E. Ruark and H. C. Urey: "Atoms, Molecules and Quanta," Chap.
XII, McGrawHill Book Company, Inc., New York, 1930. A general
discussion of the types of molecular spectra found experimentally, with the
theoretical treatment of some of them.
W. Wbizel: " Bandenspektren, " Handbuch der Experimentalphysik
(WienHarms), Erganzungsband I. A complete discussion of the theory
X38] THE ROTATION OF MOLECULES IN CRYSTALS 293
and results for diatomic molecules, with some reference to polyatomic
molecules.
D. M. Dennison: Rev. Mod. Phys. 3, 280 (1931). A discussion of the
methods of treating the rotational and vibrational spectra of polyatomic
molecules.
R. de L. Kronig: "Band Spectra and Molecular Structure,' ' Cambridge
University Press, 1930. The theory of diatomic spectra, including con
sideration of electron spins.
A. Schaefer and F. Matossi: "Das Ultrarote Spektrum," Julius
Springer, Berlin, 1930. Methods, theory, and results of infrared
spectroscopy.
K. W. F. Kohlrausch : "Der SmekalRamanEffekt," Julius Springer,
Berlin, 1931.
G. Placzek: " RayleighStreuung und Ramaneffekfc," Handbuch der
Radiologic, Vol.' VI, Akademische Verlagsgesellschaft, Leipzig, 1934.
E. Teller: "Theorie der langwelligen Molekulspektren," Hand und
Jahrbuch der chemisehen Physik, Vol. IX, Akademische Verlagsgesellschaft,
Leipzig, 1934.
CHAPTER XI
PERTURBATION THEORY INVOLVING THE TIME, THE
EMISSION AND ABSORPTION OF RADIATION, AND
THE RESONANCE PHENOMENON
39. THE TREATMENT OF A TIMEDEPENDENT PERTURBATION
BY THE METHOD OF VARIATION OF CONSTANTS
There have been developed two essentially different wave
mechanical perturbation theories. The first of these, due to
Schrodinger, provides an approximate method of calculating
energy values and wave functions for the stationary states of a
system under the influence of a constant (timeindependent)
perturbation. We have discussed this theory in Chapter VI.
The second perturbation theory, which we shall treat in the
following paragraphs, deals with the time behavior of a system
under the influence of a perturbation; it permits us to discuss such
questions as the probability of transition of the system from one
unperturbed stationary state to another as the result of the
perturbation. (In Section 40 we shall apply the theory to
the problem of the emission and absorption of radiation.) The
theory was developed by Dirac. 1 It is often called the theory
of the variation of constants; the reason for this name will be
evident from the following discussion.
Let us consider an unperturbed system with wave equation
including the time
*•*•£ IT (39  1}
the normalized general solution of which is
oo
M*> = %a n *l, (392)
n0
*P. A. M. Dirac, Proc. Roy. Soc. A 112, 661 (1926); A 114, 243 (1927).
Less general discussions were also given by Schrddinger in his fourth 1926
paper and by J. C. Slater, Proc. Nat. Acad. Set. 13, 7 (1927).
294
XI39] TIMEDEPENDENT PERTURBATION 295
in which the a n 's are constants, with ^a*a n = 1, and the
n
y„8 are the timedependent wave functions for the stationary
states, the corresponding energy values being Wo, W\, • • • ,
Wl, • • • . Now let us assume that the Hamiltonian for the
actual system contains in addition to H° (which is independent
of t) a perturbing term H r , which may be a function of the time
as well as of the coordinates of the system. 1 (For example,
H' might be zero except during the period t x < t < h, the
perturbation then being effective only during this period.)
Since we desire to express our results in terms of the unperturbed
wave functions including the time, we must consider the Schrod
inger time equation for the system. This equation is
(flf . + IT)* £.** (393)
A wave function satisfying this equation is a function of the
time and of the coordinates of the system. For a given value
of t, say t' , y?(t f ) is a function of the coordinates alone. By
the general expansion theorem of Section 22 it can be represented
as a series involving the complete set of orthogonal wave functions
for the unperturbed system,
*(x u • • • , z N , t') = 2 a »*S(*i> ''',*», O, (394)
n
the symbol ¥°(zi, • • • , Zn, t f ) indicating that t f is introduced
in place of t in the exponential time factors. The quantities
a n are constants. For any other value of t a similar expansion
can be made, involving different values of the constants a n .
A general solution of the wave equation 393 can accordingly
be written as
¥(*!, • • • , z N , t) = 2a n (0*2(*i, ••,**, 0, (395)
n
the quantities a n (t) being functions of t alone, such as to cause SP
to satisfy the wave equation 393.
The nature of these functions is found by substituting the
expression 395 in the wave equation 393, which gives
1 W might also be a function of the momenta p xlt • • *• , which would then
be replaced by ■— — , ....
2irt dxi
296 PERTURBATION THEORY INVOLVING THE TIME [XI 39a
n n
The first and last terms in this equation cancel (by Eqs. 391 and
392), leaving
n n
If we now multiply by ^* and integrate over configuration space,
noting that all terms on the left vanish except that for n = m
because of the orthogonality properties of the wave functions,
we obtain
oo
d m {t) = ^^a n (t)f* Q m *H'*°d Ty
m = 0, 1, 2,
(396)
This is a set of simultaneous differential equations in the functions
a,m{t), by means of which these functions can be evaluated in
particular cases.
39a. A Simple Example. — As an illustration of the use of the
set of equations 396, let us consider that at the time t =
we know that a system in which we are interested is in a particular
stationary state, our knowledge perhaps having been obtained
by a measurement of the energy of the system. The wave func
tion representing the system is then ^°, in which I has a particular
value. If a small perturbation H ' acts on the system for a short
time t f , H r being independent of t during this period, we may
solve the equations 396 by neglecting all terms on the right side
except that with n = I) that is, by assuming that only the term
in ai(t) need be retained on the right side of these equations.
It is first necessary for us to discuss the equation for aj itself.
This equation (Eq. 396 with m = I and a n = for n 5* I) is
dT~ —T ai{t)Hll >
in which H' u = f\l/i*H'\l/idT, which can be integrated at once to
give
ai (t) = e~ 2 * iH ii t/h } ^t ^t', (397)
XI39a] TIMEDEPENDENT PERTURBATION 297
the assumption being made that a* = 1 at the time t = 0.
This expression shows the way that the coefficient ai changes
during the time that the perturbation is acting. During this
time the wave function, neglecting the terms with m ?± l } is
a,Wl = ife^'^'
It will be observed that the timedependent factor contains the
firstorder energy W° + H[ u as given by the Schrodinger per
turbation theory; this illustrates the intimate relation of the
two perturbation theories.
Now let us consider the remaining equations of the set 396,
determining the behavior of the coefficients a m (t) with m 5* I.
Replacing ai on the right side of 396 by its initial value 1
ai(0) = 1, and neglecting all other a n 's, we obtain the set of
approximate equations
da m (t) 2wi
 f ¥£*#'*
dt ~ h ■"'*'*
This can be written as
da m (t) = ~™H' ml e * dt, <^t ^t',m*l,
in which
H' ml = WH'tfdr, (398)
and H' m i is independent of t, since we have considered W to be
independent of / during the period <J / ^ V , and have replaced
the timecontaining wave functions Mf£* an d ^? by the amplitude
functions \[/%* and \pi and the corresponding time factors, the
latter being now represented explicitly by the exponential
functions. These equations can be integrated at once; on intro
ducing the limits, and noting that a m (0) =0 for m t*1 % we
obtain
2iri(W m  Wi)t'
o.(0 = H'J ~^ m _ h Wi , m*l, (399)
in which, it is remembered, the subscript I refers to the state
initially occupied and m to other states. In case that the time
V is small compared with the time h/(W m — Wi) y the expression
can be expanded, giving
1 The expression for ai(t) given by Equation 397 could be introduced in
place of gu(0) ~ 1, with, however, no essential improvement in the result.
298 PERTURBATION THEORY INVOLVING THE TIME [XI39a
ajfi « X^^' m * l (3& " 10)
At the time t' the wave function for the system (which was
tyf at time t = 0) is approximately
*(*') = ai (t')*f + %'a m (0*°n> (3911)
m
(the prime on the summation sign indicating that the term
m = I is omitted), in which aj is nearly equal to 1 and the a m 's
are very small. This wave function continues to represent the
system at times later than t', so long as it remains isolated. We
could now carry out a measurement (of the energy, say) to deter
mine the stationary state of the system. The probability that
the system would be found in the rath stationary state is a£a m .
This statement requires the extension of the postulates regarding the
physical interpretation of the wave equation given in Sections 10c and I2d.
It was shown there that an average value could be predicted for a dynamical
function for a system at time t from a knowledge of the wave function
representing the system. The average value predicted for the energy of a
system with wave function ¥ = 2^.a n *l is W = 2^cL* a nWl. However, an
n n
actual individual measurement of the energy must give one of the values
Wq, W\, W% etc., inasmuch as it is only for wave functions corresponding
to stationary states that the energy has a definite value (Sec. 10c). Hence
when a measurement of the energy has been made, the wave function
representing the system is no longer V = 2 an ^°> DU ^ ^ one °* tne functions
*J, *J, *?, etc.
This shows how a wave function does not really represent the system
but rather our knowledge of the system. At time t — we knew the energy
of the system to be TFJ, and hence we write ^t\ for the wave function. (We
do not know everything about the system, however; thus we do not know
the configuration of the system but only the probability distribution func
tion ¥j*¥{.) At time V we know that at time t = the wave function was
¥j, and that the perturbation H' was acting between times t — and V.
From this information we obtain the wave function of Equations 3911,
3910, and 398 as representing our knowledge of the system. With it we
predict that the probability that the system is in the mth stationary state
is a*a m . So long as we leave the system isolated, this wave function repre
sents the system. If we allow the system to be affected by a known pertur
bation, we can find a new wave function by the foregoing methods. If we
now further perturb the system by an unknown amount in the process of
making a measurement of the energy, we can no longer apply these methods;
XI40a] EMISSION AND ABSORPTION OF RADIATION 299
instead, we assign to the system a new wave function compatible with our
new knowledge of the result of the experiment.
A more detailed discussion of these points will be given in Chapter XV.
Equation 3910 shows that in case t' is small the probability of
finding the system in the stationary state m as a result of transi
tion from the original state I is
a:a m = ^H'^H' ml t", (3912)
being thus proportional to the square of the time t r rather than
to the first power as might have been expected. In most cases
the nature of the system is such that experiments can be designed
to measure not the probability of transition to a single state but
rather the integrated probability of transition to a group of
adjacent states; it is found on carrying out the solution of the
fundamental equations 396 and subsequent integration that for
small values of t' the integrated probability of transition is pro
portional to the first power of the time t'. An example of a
calculation of a related type will be given in Section 406.
40. THE EMISSION AND ABSORPTION OF RADIATION
Inasmuch as a thoroughly satisfactory quantummechanical
theory of systems containing radiation as well as matter has not
yet been developed, we must base our discussion of the emission
and absorption of radiation by atoms and molecules on an
approximate method of treatment, drawing upon classical electro
magnetic theory for aid. The most satisfactory treatment of
this type is that of Dirac, 1 which leads directly to the formulas
for spontaneous emission as well as absorption and induced
emission of radiation. Because of the complexity of this theory,
however, we shall give a simpler one, in which only absorption
and induced emission are treated, prefacing this by a general
discussion of the Einstein coefficients of emission and absorption
of radiation in order to show the relation that spontaneous
emission bears to the other two phenomena.
40a. The Einstein Transition Probabilities. — According to
classical electromagnetic theory, a system of accelerated electri
cally charged particles emits radiant energy. In a bath of
*P. A. M. Dirac, Proc. Roy. Soc. A112, 661 (1926); A114, 243 (1927);
J. C. Slater, Proc. Nat. Acad. Set. 13, 7 (1927).
300 PERTURBATION THEORY INVOLVING THE TIME [XI40a
radiation at temperature T it also absorbs radiant energy, the
Tates of absorption and of emission being given by the classical
laws. These opposing processes might be expected to lead to a
state of equilibrium. The following treatment of the correspond
ing problem for quantized systems (atoms or molecules) was
given by Einstein 1 in 1916.
Let us consider two nondegenerate stationary states m and n
of a system, with energy values W m and W n such that W m is
greater than W n . According to the Bohr frequency rule, transi
tion from one state to another will be accompanied by the
emission or absorption of radiation of frequency
_ W m  Wn
Vmn ~ h
We assume that the system is in the lower state n in a bath of
radiation of density p(y mn ) in this frequency region (the energy
of radiation between frequencies v and v + dv in unit volume
being p{v)dv). The probability that it will absorb a quantum
of energy of radiation and undergo transition to the upper state
in unit time is
B n +mp(Vmn)
B n * m is called Einstein's coefficient of absorption. The proba
bility of absorption of radiation is thus assumed to be propor
tional to the density of radiation. On the other hand, it is
necessary in order to carry through the following argument to
postulate 2 that the probability of emission is the sum of two
parts, one of which is independent of the radiation density and
the other proportional to it. We therefore assume that the
probability that the system in the upper state m will undergo
transition to the lower state with the emission of radiant energy
is
A m ~*n ~T ■t>m~>nP\Vmn)»
A m >n is Einstein 7 s coefficient of spontaneous emission and 2? m > n
is Einstein 9 s coefficient of induced emission.
1 A. Einstein, Verh. d. Deutsch. Phijs. Ges. 18, 318 (1916); Phys. Z. 18,
121 (1917).
2 This postulate is of course closely analogous to the classical theory,
according to which an oscillator interacting with an electromagnetic wave
could either absorb energy from the field or lose energy to it, depending
on the relative phases of oscillator and wave.
XI40a] EMISSION AND ABSORPTION OF RADIATION 301
We now consider a large number of identical systems of this
type in equilibrium with radiation at temperature T. The
density of radiant energy is known to be given by Planck's
radiation law as
p(v) = r  5  , (401)
e kf  1
in which k is the Boltzmann constant. Let the number of sys
tems in state m be N m , and that in state n be N n . The number of
systems undergoing transition in unit time from state n to state
m is then
N n Bn+mp(v mn ) y
and the number undergoing the reverse transition is
N m {A m +n + B m > n p(v mn )}.
At equilibrium these two numbers are equal, giving
iVffi & n +mP\ymn)
The equations of quantum statistical mechanics (Sec. 49)
require that the ratio N n /N m be given by the equation
ip = e kT = e h "»» /kT . (403)
From Equations 402 and 403 we find for p(v mn ) the expression
A _>
p( v ^) = £ ^J£"_. B ( 40 ~4)
'm—m
In order for this to be identical with Equation 401, we must
assume that the three Einstein coefficients are related by the
equations
B n >m = B m +n (40"6o)
and
A m ^ n = ^^B m >n; (4056)
that is, the coefficients of absorption and induced emission are
equal and the coefficient of spontaneous emission 1 differs from
them by the factor fhchv^Jc*.
1 It is interesting to note that at the temperature T = •— —  the proba
k log 2
bilities of spontaneous emission and induced emission are equal.
302 PERTURBATION THEORY INVOLVING THE TIME [XI40b
40b. The Calculation of the Einstein Transition Probabilities
by Perturbation Theory. — According to classical electromagnetic
theory, the density of energy of radiation of frequency v in space,
with unit dielectric constant and magnetic permeability, is
given by the expression
P« = ^TO, (40*)
in which E\v) represents the average value of the square of the
electric field strength corresponding to this radiation. The
distribution of radiation being isotropic, we can write
%W$) = ^ = TO = Wy), (407)
E x {y) representing the component of the electric field in the
x direction, etc. We may conveniently introduce the time
variation of the radiation by writing
E x (v) = 2E° x (v) cos 2wvt = E° x (v)(e** M + e*™'), (408)
the complex exponential form being particularly convenient for
calculation. Since the average value of cos 2 2wvt is J^, we see
that
pW = ±W(v) = I'm*) = £tf."«. (409)
Let us now consider two stationary states m and n of an
unperturbed system, represented by the wave functions ^ and
^£, and such that W m is greater than W n . Let us assume that
at the time t = the system is in the state n, and that at this
time the system comes under the perturbing influence of radiation
of a range of frequencies in the neighborhood of v mn , the electric
field strength for each frequency being given by Equation 408.
We shall calculate the probability of transition to the state m
as a result of this perturbation, using the method of Section 39.
The perturbation energy for a system of electrically charged
particles in an electric field E x parallel to the x axis is
W = E x ^x u (4010)
in which e, represents the charge and x, the x coordinate of the
jth particle of the system. The expression ]xe;£, (the sum being
XI40b] EMISSION AND ABSORPTION OF RADIATION 303
taken over all particles in the system) is called the component of
electric dipole moment of the system along the x axis and is often
represented by the symbol /**. We now make the approximation
that the dimensions of the entire system (a molecule, say) are
small compared with the wave length of the radiation, so that
the electric field of the radiation may be considered constant over
the system. In the case under consideration the field strength E x
is given by the expression
E x = $El{v)(e 2 * ivt + e 2 ™ l )dv.
Let us temporarily consider the perturbation as due to a single
frequency v. Introducing a m (0) = and a n (0) = 1 in the right
side of Equation 396 (a n being the coefficient of a particular
state and all the other coefficients in the sum being zero), this
equation becomes
MO = ~f*l*H'*°dr  ~ji:*e^ W ^ x ( v )(e^> +
e 2*ivt} J^ qxtfle h " dr.
If we now introduce the symbol n Zmn to represent the integral
M* W n = fK*Xe 3 xMdr = fr m W»dr, (4011)
we obtain the equation
da m (t)
dt
9 i ( 2ir1
'(WmWn
> + kv)t
+
2xi
e h
(W n 
which gives,
on in
tegration,
'1 .T^
■Wn+K*)t
a m (t) = n Xmn El(v)\
1 1 — e n
+ hv
+
1 
e h
Wn
hp)i*
W»hp)t\
r
W m ~ Wn
(4012)
Of the two terms of Equation 4012, only one is important,
and that one only if the frequency v happens to lie close to
Vmn = (W m — W n )/h. The numerator in each fraction can vary
in absolute magnitude only between and 2, and, inasmuch as
for a single frequency the term n Xmn El(v) is always small, the
304 PERTURBATION THEORY INVOLVING THE TIME [XI40b
expression will be small unless the denominator is also very
small; that is, unless hv is approximately equal to W m — W n .
In other words, the presence of the socalled resonance denomina
tor Wm — Wn — hv causes the influence of the perturbation
in changing the system from the state n to the higher state m
to be large only when the frequency of the light is close to that
given by the Bohr frequency rule. In this case of absorption, it
is the second term which is important; for induced emission of
radiation (with W m — W n negative), the first term would play
the same role.
Neglecting the first term, we obtain for a*(t)a m (t), after slight
rearrangement, the expression
SUlt&Wm  Wn kv)t\
a*(t)a m {t) = 4( M J% 02 (,) ( j
(W m ~Wn hvY
(If ji Xmn is complex, the square of its absolute value is to be used
in this equation.) This expression, however, includes only the
terms due to a single frequency. In practice we deal always
with a range of frequencies. It is found, on carrying through
the treatment, that the effects of light of different frequencies are
additive, so that we now need only to integrate the above
expression over the range of frequencies concerned. The
integrand is seen to make a significant contribution only over the
region of v near v mny so that we are justified in replacing El(v)
by the constant El(v mn ), obtaining
a*(t)a m (t) = 4U ! ^ ! ('..)
/sin 2 t(W m W n  hv)t\
(W m W n  hvY °
This integral can be taken from — °o to + °° , inasmuch as the
value of the integrand is very small except in one region; and
I sin^ x
making use of the relation I — ^— dx = t, we can obtain the
J— » #
equation
a*(t)a m (t) = ^0O^2 f (O* (4013)
It is seen that, as the result of the integration over a range of
values of v } the probability of transition to the state m in time t
is proportional to t f the coefficient being the transition probability
S340b] EMISSION AND ABSORPTION OF RADIATION 305
as usually defined. With the use of Equation 409 we may now
introduce the energy density p(v mn ), obtaining as the probability
of transition in unit time from state n to state m under the
influence of radiation polarized in the x direction the expression
3^2(Mx m n) 2 p(^mn).
The expressions for the y and z directions are similar. Thus we
obtain for the Einstein coefficient of absorption B n > m the
equation
B n ^ m = g{ ( M ,„„) 2 + GO' + W !  (4014a)
By a completely analogous treatment in which the values
«n(0) = 0, a m (0) = 1 are used, the Einstein coefficient of induced
emission B m > n is found to be given by the equation
B m ^ n = p{(Mx.J 2 + U.J 2 + U.n) 2 }, (40146)
as, indeed, is required by Equation 405a.
Our treatment does not include the phenomenon of spontaneous
emission of radiation. Its extension to include this is not easy;
Dirac's treatment is reasonably satisfactory, and we may hope
that the efforts of theoretical physicists will soon provide us with
a thoroughly satisfactory discussion of radiation. For the
present we content ourselves with using Equation 4056 in
combination with the above equations to obtain
A m ^ n = ijgk{0O« + (*..)* + GO*} (4015)
as the equation for the Einstein coefficient of spontaneous
emission.
As a result of the foregoing considerations, the wavemechanical
calculation of the intensities of spectral lines and the determina
tion of selection rules are reduced to the consideration of the
electricmoment integrals defined in Equation 4011. We shall
discuss the results for special problems in the following sections.
It is interesting to compare Equation 4015 with the classical
expression given by Equation 34 of Chapter I. Recalling that
the energy change associated with each transition is kv mn , we
see that the wavemechanical expression is to be correlated with
306 PERTURBATION THEORY INVOLVING THE TIME [XI40d
the classical expression for the special case of the harmonic
oscillator by interpreting Mx™ as onehalf the maximum value
exo of the electric moment of the classical oscillator.
40c. Selection Rules and Intensities for the Harmonic Oscil
lator. — The electric dipole moment for a particle with electric
charge e carrying out harmonic oscillational motion along the
x axis (a neutralizing charge —e being at the origin) has the
components ex along this axis and zero along the y and z axes.
The only nonvanishing dipole moment integrals /**•« = &c m n
have been shown in Section lie to be those with m = n + 1 or
m = n — 1. Hence the only transitions which this system can
undergo with the emission or absorption of radiation are those
from a given stationary state to the two adjacent states. 1 The
selection rule for the harmonic oscillator is therefore An = ±1,
and the only frequency of light emitted or absorbed is v . The
expression for z n ,ni in Equation ll25a corresponds to the
value
with a — 4v 2 mvo/h y for the coefficient of spontaneous emission,
with similar expressions for the other coefficients. An applica
tion of this formula will be given in Section 40e.
Problem 401. Show that for large values of n Equation 4016 reduces
to the classical expression for the same energy.
Problem 402. Discuss the selection rules and intensities for the three
dimensional harmonic oscillator with characteristic frequencies v Xi vy 9
and v t .
Problem 403. Using firstorder perturbation theory, find perturbed
wave functions for the anharmonic oscillator with V = 2ic i mvlx 2 +■ ax 3 ,
and with them discuss selection rules and transition probabilities.
40d. Selection Rules and Intensities for Surface harmonic
Wave Functions. — In Section 18 we showed that the wave
functions for a system of two particles interacting with one
another in the way corresponding to the potential function
1 This statement is true only to within the degree of approximation of our
treatment. A more complete discussion shows that transitions may also
occur as a result of interactions corresponding to quadrupole terms and still
higher terms, as mentioned in Section 3, and as a result of magnetic
interactions.
XI40d] EMISSION AND ABSORPTION OF RADIATION 307
V(r), in which r is the distance between the two particles, are
of the form
in which the &,<p functions Bim(&)$m(<p) are surface harmonics,
independent of V(r). We can hence discuss selection rules and
intensities in their dependence on I and m for all systems of
this type at one time.
The components of electric dipole moment along the x, y,
and z axes are
Mx = n(r) sin # cos <p,
Mi/ = n(r) sin # sin <p,
and
\x t = /x(r) cos #,
in which n(r) is a function of r alone, being equal to er for two
particles with charges +e and — e. Each of the dipole moment
integrals, such as
"..„.„.„„.. = ///BSWe,* (*)«(^) M (r) sin # cos *>
i2*'j<(r)6rm' (#)<l> m ^)r 2 sin #d<pd#dr,
can accordingly be written as the product of three factors, one
involving the integral in r, one the integral in #, and one the
integral in <p:
»V n i^l> m , = M»i»'i/» tal , w ,fo_,,> (4017)
in which
Hnin>i> = J[ °°^(r) M (r)i? n ,Kr)r 2 dr, (4018)
/v — 7 = I e 'W 8in *>Qi'*'W sin &», (4019)
'*— ') ^2 ( COS *>)
*'[ = X T ^ ( ^) sin ^*«'W^ ( 4 ° 2 °)
(In Equations 4019 and 4020 the subscripts x, y, and z are
respectively associated with the three factors in braces.)
Let us first discuss the light polarized along the z axis, cor
responding to the dipole moment n t . From the orthogonality
and
308 PERTURBATION THEORY INVOLVING THE TIME [XI40d
and normalization integrals for $(<p) we see that
Qz = for m! 7* m
° mm'
and
Qtmm = 1.
In discussing /* we consequently need consider only the
integrals with m' = m. It is found with the use of the recursion
formula (Prob. 192)
COS tfP[">(COS 0) = ^+Mp lml i(cog 0) +
 ~{2lt 1) 1) p '"' l(cOS *> (40_21)
that fz lmVm , vanishes except when V is equal to I + 1 or I — 1.
A similar treatment of the integrals for x and 2/ shows that
light polarized along these axes is emitted only when m changes
by +1 or — 1, and I changes by +1 or — 1.
We have thus obtained the selection rules Am = 0, +1, or —1
and AZ = +1 or — 1. The selection rule for I is discussed in
the following sections. That for m can be verified experimentally
only by removing the degeneracy, as by the application of a
magnetic field; it is found, in agreement with the theory, that
in the Zeeman effect the light corresponding to Am = is
polarized along the z axis (the axis of the magnetic field), and that
corresponding to Am = ± 1 is polarized in the xy plane.
The values of the products of the factors / and g are
(M  i(fiis  i/(* + MX* + M  D l* \
WJ W».ii»  %ug)v  2\ (21 + l)(2l  1) j \
m  i(fQ) l/ (^ H)g  H + 1) H (4022)
_ / (i + M)(Mm) y*
utf; ^ii^M«i  \ (a + l)(2l  1) /
with similar expressions for the transitions I to I + 1, etc.
Problem 404. Using Equation 4021, obtain selection rules and
intensities for ju*.
Problem 406. Similarly derive the other formulas of 4022.
Problem 406. Calculate the total probability of transition from one
level with given value of / to another, by summing over m. By separate
summation for /x x , n Vf and m* show that the intensity of light polarized along
these axes is the same.
XI40e] EMISSION AND ABSORPTION OF RADIATION 309
40e. Selection Rules and Intensities for the Diatomic Mole
cule. The Franck Condon Principle. — A very simple treatment
of the emission and absorption of radiation for the diatomic
molecule can be given, based on the approximate wave functions
of Section 35c. For the complex system of two nuclei and several
electrons the electric dipole moment n(r) can be expanded as a
series in r — r ,
M(r) = mo + €(r  r ) + • • • , (4023)
in which e is a constant. The permanent dipole moment /xo is
the quantity which enters in the theory of the dielectric constant
of dipole molecules; its value is known from dielectric constant
measurements for many substances.
Introducing this expansion in Equation 4018, we find as a
first approximation that n may change by or by ±1. In the
former case the emission or absorption of radiation is due to
the constant term fi , and in the latter case to the term e(r — r ),
the integrals being then similar to the harmonic oscillator
integrals. The values of ju nn / are
linn = mo (4024a)
and
Mn.n1  t^f (40246)
in which a = ^T 2 fxv /h (/x being here the reduced mass for the
molecule). The selection rules and intensity factors for I and
m are as given in the preceding section.
It is found experimentally that dipole molecules such as the
hydrogen halides absorb and emit pure rotation and oscillation
rotation bands in accordance with these equations. In all
these bands the selection rule Al = ± 1 is obeyed, and Zeeman
effect measurements have shown similar agreement with the
selection rule for m. The intensities of lines in the pure rotation
bands show rough quantitative agreement with Equation
4024a, using the dielectric constant value of /* , although because
of experimental difficulties in the far infrared the data are as
yet not very reliable. Measurements of absorption intensities
for An = 1 have been used to calculate c As seen from the
following table, « is of the order of magnitude of mo/Vo, so that
these molecules may be considered roughly as equivalent to two
particles of charges +€ and — €.
310 PERTURBATION THEORY INVOLVING THE TIME [XI 40©
Table 401
Mo (dielectric constant)
r
»o/r
HC1
HBr
HI
1.034 X 10 18 e.s.u.
0.788
0.382
1.28 A
1.42
1.62
0.169 e
.116
.049
0.086 e
.075
.033
i E. BabtholomjS, Z. phya. Chem. B 23, 131 (1933).
It is also observed that oscillationrotation bands with
An = 2, 3, etc. occur; this is to be correlated with the deviation
of the potential function V(r) from a simple quadratic function.
In the foregoing discussion we have assumed the electronic
state of the molecule to be unchanged by the transitions. The
selection rule for n and the intensities are different in case there is a
change in the electronic state, being then determined, according
to the FranckCondon principle, 1 mainly by the nature of the
electronic potential functions for the two electronic states. As
we have seen in Section 34, there is little interaction between the
electronic motion and the nuclear motion in a molecule, and
during an electronic transition the internuclear distance and
nuclear velocities will not change very much. Let us consider
the two electronic states A and B, represented by the potential
curves of Figure 401, in which the oscillational levels are also
shown. If the molecule is in the lowest oscillational level
n' = of the upper state, the probability distribution function
for r is large only for r close to r v We would then expect an
electronic transition to state B to leave the molecule at about the
point Pi on the potential curve, the nuclei having only small
kinetic energy; these conditions correspond to the levels n" = 7
or 8 for the lower state.
This simple argument is justified by wavemechanical con
siderations. Let us consider that the wave functions for the
upper electronic state may be written as &>$»', in which fa*
represents the nuclear oscillational part of the wave function
described by the quantum number n', and ty a > the rest of the wave
function (electronic and nuclear rotational), the symbol or'
representing all other quantum numbers. Similarly, we write
^r"^n" for the wave functions for the lower electronic state.
1 J. Fbanck, Trans. Faraday Soc, 21, 536 (1926); E. U. Condon, Phys
Rev. 28, 1182 (1926); 32, 868 (1928).
XI40e] EMISSION AND ABSORPTION OF RADIATION 311
The electric dipole moment integrals /**,,* v»"> *V*w> an( *
H s , nV , M „ are of the form
^Vnw = /^VnVx^^ndr. (4025)
We assume that in this case, when there is a change in the
w
r eA
r AB
Fig. 401. — Energy curves for two electronic states of a molecule, to illustrate
the FranckCondon principle.
electronic state, the dipole moment function /* changes only
slowly with change in the internuclear separation r, being deter
mined essentially by the electronic coordinates. Neglecting the
312 PERTURBATION THEORY INVOLVING THE TIME [XI40f
dependence of /z on r, we can then integrate over all coordinates
except r, obtaining
*V.w = M v JW»rWr. (4026)
The integral in r, determining the relative intensities of the
various n'n" bands, is seen to have the form of an orthogonality
integral in r. Hence if the two potential functions Va and V B
were identical except for an additive constant the integral would
vanish except for n' = n", the selection rule for n then being
An = o. In the case represented by Figure 401, the wave func
tion i/v with n' = is large only in the neighborhood of r = r ejL .
The wave functions i> n >> with n" = 7 or 8 have large values in
this region, so that the bands n f = — > n" — 7 or 8 will be
strong. The intensity of the bands for smaller or larger values
of n" will fall off. For smaller values of n" the functions i/v
show the rapid exponential decrease in the region near r e ^
(corresponding to the fact that the classical motion of the nuclei
would not extend into this region); whereas for larger values
of n" the functions \p n >> show a rapid oscillation between positive
and negative values, causing the integral with the positive
function yf/ n > with n r — to be small (the oscillation of ^v
between positive and negative values corresponding to large
nuclear velocities in the classical motion).
Similarly the transitions from the level with n' = 5, the wave
function for which has its maximum values near the points
P 2 and P 3 , will occur mainly to the levels n" = 2 or 3 and
n" = 11 or 12. 1
40f . Selection Rules and Intensities for the Hydrogen Atom. —
The selection rule for Z, discussed in Section 40d, allows only
transitions with AZ = ± 1 for the hydrogen atom. The lines of
the Lyman series, with lower state that with n — 1 and I = 0,
are in consequence due to transitions from upper states with
1 = 1. The radial electric dipole moment integral
Hn> n >> = $R* v {r)rR n >>i"{r)rHr
has been evaluated by Pauli 2 for several special cases. For
1 For a more complete discussion of this subject the reader is referred to
the papers of Condon and to the discussions in Condon and Morse, " Quan
tum Mechanics/' Chap. V, and Ruark and Urey, "Atoms, Molecules and
Quanta," Chap. XII.
2 Communicated in Schrddinger's third 1926 paper.
XI40g] EMISSION AND ABSORPTION OF RADIATION 313
n" = 1, V = its value corresponds to the total intensity,
aside from a constant factor, of
in'l =
n'(n' + 1) 2 » +1
This has a non vanishing value for all values of n 1 greater than 1.
Hence there is no selection rule for n for the Lyman series, all
transitions being allowed. It is similarly found that there is no
selection rule for n for spectral series in general.
For the Balmer series, with lower state that with n = 2 and
I = or 1, the selection rule for I permits the transitions — > 1,
1 — >0, and 2 — > 1. The total intensity corresponding to these
transitions from the level n = n f to n = 2 is, except for a constant
factor,
7 "  Mrr^ i3n ' 2 ~ 4)(5n ' 2 " 4) 
The operation of the selection rule for I for hydrogen and
hydrogenlike ions can be seen by the study of the fine structure
of the lines. The phenomena are complicated, however, by the
influence of electron spin. 1 In alkali atoms the levels with given
n and varying I are widely separated, and the selection rule for I
plays an important part in determining the nature of their
spectra. Theoretical calculations have also been made of the
intensities of lines in these spectra with the use of wave functions
such as those described in Chapter IX, leading to results in
approximate agreement with experiment.
40g. Even and Odd Electronic States and Their Selection
Rules. — The wave functions for an atom can all be classified as
either even or odd. An even wave function of N electrons is
one such that \p(xi, y u Zi, x^ • • • , Zn) is equal to \p(— Xi, —y u
— Zi,—X2,'*', — s>r); that is, the wave function is unchanged
on changing the signs of all of the positional coordinates of the
electrons. An odd wave function is one such that t(zi, yi,
ti,X2,   • , zn) is equal to ^(Xi, y lf z h —x 2 , • • • , —Zn).
Now we can show that the only transitions accompanied by the
emission or absorption of dipole radiation which can occur are
those between an even and an odd state (an even state being one
represented by an even wave function, etc.). The electric
1 See Pauling and Goudsmit, "The Structure of Line Spectra," Sec. 16.
314 PERTURBATION THEORY INVOLVING THE TIME [XI41
N
moment component functions £\ex if etc. change sign in case
that the electronic coordinates are replaced by their negatives.
Consequently an electricmoment integral such as ^t'^ex^n^dr
i
will vanish in case that both ^v and ^ n " are either even or odd,
but it is not required to vanish in case that one is even and the
other odd. We thus have derived the very important selection
rule that transitions with the emission or absorption of dipole
radiation are allowed only between even and odd states. Because of
the practical importance of this selection rule, it is customary to
distinguish between even and odd states in the term symbol, by
adding a superscript ° for odd states. Thus various even states
are written as X >S, 3 P, 2 D, etc., and odd states as 1 5°, 3 P°, 2 D°, etc.
In case that the electronic configuration underlying a state is
known, the state can be recognized as even or odd. The one
electron wave functions are even for I = 0, 2, 4, etc. (s, d, g, etc.,
orbitals) and odd for I = 1, 3, 5, etc. (p, /, h y etc., orbitals).
Hence the configuration leads to odd states if it contains an
odd number of electrons in orbitals with I odd, and otherwise to
even states. For example, the configuration ls 2 2s 2 2p 2 leads to the
even states 1 S ) l D, and 3 P, and the configuration ls 2 2p3d to
the odd states »P°, l D°, l F° y 3 P°, 3 D°, and 3 P°.
Even and odd states also occur for molecules, and the selection
rule is also valid here. A further discussion of this point will be
given in Section 48.
Problem 407. Show that the selection rules forbid a hydrogen atom
in a rectangular box to radiate its translational kinetic energy. Extend the
proof to any atom in any kind of box.
41. THE RESONANCE PHENOMENON
The concept of resonance played an important part in the dis
cussion of the behavior of certain systems by the methods of
classical mechanics. Very shortly after the discovery of the
new quantum mechanics it was noticed by Heisenberg that a
quantummechanical treatment analogous to the classical
treatment of resonating system can be applied to many problems,
and that the results of the quantummechanical discussion in these
cases can be given a simple interpretation as corresponding to a
quantummechanical resonance phenomenon. It is not required
XI 41a]
THE RESONANCE PHENOMENON
315
that this interpretation be made; it has been found, however,
that it is a very valuable aid to the student in the development
of a reliable and productive intuitive understanding of the
equations of quantum mechanics and the results of their applica
tion. In the following sections we shall discuss first classical
resonance and then resonance in quantum mechanics.
41a. Resonance in Classical Mechanics. — A striking phe
nomenon is shown by a classical mechanical system consisting of
two parts between which there is operative a small interaction,
the two parts being capable of executing harmonic oscillations
with the same or nearly the same frequency. It is observed
that the total oscillational energy fluctuates back and forth
Fig. 411.
A system of coupled pendulums, illustrating the phenomenon of
resonance.
between the two parts, one of which at a given time may be
oscillating with large amplitude, and at a later time with small
amplitude, while the second part has changed jn the opposite
direction. It is customary to say that the two parts of the
system are resonating. A familiar example of such a system is
composed of two similar tuning forks attached to the same base.
After one fork is struck, it gradually ceases to oscillate, while at
the same time the other begins its oscillation. Another example
is two similar pendulums connected by a weak spring, or attached
to a common support in such a way that interaction of the two
occurs by way of the support (Fig. 411). It is observed that if
only one pendulum is set to oscillating, it will gradually die down
and stop, while the other begins to oscillate, ultimately reaching
the amplitude of oscillation initially given the first (neglecting the
(412)
316 PERTURBATION THEORY INVOLVING THE TIME [XI41a
f rictional dissipation of energy) ; and that this process of transfer
of energy from one pendulum to the other is repeated over and
over.
It is illuminating to consider this system in greater detail.
Let Zi and x 2 be the coordinates for two oscillating particles each
of mass m (such as the bobs of two pendulums restricted to
small amplitudes, in order that their motion be harmonic),
and let v be their oscillational frequency. We assume for the
total potential energy of the system the expression
V(xi, x 2 ) = 2K <l mv\x\ + 2w 1 mv\x\ + 4tT 2 m\x 1 x 2j (411)
in which 4tTr 2 m\xiX2 represents the interaction of the two oscil
lators. This simple form corresponds to a Hookedlaw type of
interaction. The solution of the equations of motion is easily
accomplished by introducing the new variables 1
£ = T/^ Xl +*2),J
In terms of these, the potential energy becomes
F(£, v ) = Wm{v\ + \)e + 2*hn{v\  X)t? 2 ,
while the kinetic energy has the form
T = V 2 mx\ + }/ 2 mxl = y 2 m¥ + y 2 mi)\
These expressions correspond to pure harmonic oscillation of the
two variables £ and r\ (Sec. la), each oscillating with constant
amplitude, £ with the frequency \/v\ + X and r) with the
frequency y/ v% — X, according to the equations
£ = £0 cos {2iry/vl + X t + bz)\ (413)
7] = Tjo cos (2ttV^  X t + 5,). ) V
From these equations we obtain the equations
Xi = ^ cos {2W»l + X t) + 5?= cos (27rV^T =r X"0,)
Y 3 V2 (414)
x 2 = ~ cos (fcrVVj + X t)  52= cos (2ttv^§  X 0,1
1 These are the normal coordinates of the system, discussed in Section 37.
XI41a] THE RESONANCE PHENOMENON 317
for xi and x 2 , in which we have put the phase constants 8$ and 8,
equal to zero, as this does not involve any loss in generality
It is seen that for \/vl small the two cosine functions differ only
slightly from one another and both x\ and x 2 carry out approxi
mate harmonic oscillation with the approximate frequency vq,
but with amplitudes which change slowly with the time. Thus
at t ^ the cosine terms are in phase, so that x\ oscillates with
the amplitude (£ + *7o)/\/2 and X2 with the smaller amplitude
(£o — i/o)/v / 2 At the later time t = h such that
the cosine terms are just out of phase, Xi then oscillating with the
amplitude (£ — rj )/\/2 and x 2 with the amplitude (£ + ^7o)/v / 2
Thus we see that the period r of the resonance, the time required
for xi to change from its maximum to its minimum amplitude
and then back to the maximum, is given by the equation
V^f+Xr = a/^TXt + 1
or
1 ~ "o
(41~5)
It is also seen that the magnitude of the resonance depends on
the constants of integration £o and rj , the amplitudes of motion
of Xi and x 2 varying between the limits \/2£o and in case that
Vo = £o, and retaining the constant value £o/\/2 (no resonance!)
in case that r) = 0.
The behavior of the variables x± and x 2 may perhaps be
followed more clearly by expanding the radicals y/v\ + X and
V^o — X in powers of \/v% and neglecting terms beyond the
first power. After simple transformations, the expressions
obtained are
Xl = (&±Jo) cos27r^ cos 27rvot  iksiJ!S> s in 2tt* sin 2™rf
V2 "o y/2 v o
and
X2 = ^°~ 71o) cos27rtcos2irv t  ( *° + no) sin 2tt* sin 2™<>*.
V2 ^o V2 "o
It is clear from this treatment that we speak of resonance only
because it is convenient for us to retain the coordinates Xi and
318 PERTURBATION THEORY INVOLVING THE TIME (XI41b
Xi in the description of the system; that is, to speak of the motion
of the pendulums individually rather than of the system as a
whole. We can conceive of an arrangement of levers whereby
an indicator in an adjacent room would register values of £, and
another values of 77. An observer in this room would say that
the system was composed of two independent harmonic oscillators
with different frequencies and constant amplitudes, and would
not mention resonance at all.
Despite the fact that we are not required to introduce it, the
concept of resonance in classical mechanical systems has been
found to be very useful in the description of the motion of sys
tems which are for some reason or other conveniently described
as containing interacting harmonic oscillators. It is found
that a similar state of affairs exists in quantum mechanics.
Quantummechanical systems which are conveniently considered
to show resonance occur much more often, however, than
resonating classical systems, and the resonance phenomenon
has come to play an especially important part in the applications
of quantum mechanics to chemistry.
41b. Resonance in Quantum Mechanics. — In order to illus
trate the resonance phenomenon in quantum mechanics, let us
continue to discuss the system of interacting harmonic oscil
lators. 1 Using the potential function of Equation 411, the wave
equation can be at once separated in the coordinates £ and 17
and solved in terms of the Hermite functions. The energy
levels are given by the expression
W n ^ = (n* + M)hV~vJ+\ + (n, + M)hVpf=\, (416)
which for X small reduces to the. approximate expression
F,^ S (n + 1)A„ + (n«  O^  ^^! + . . . ,
(417)
in which n = n^ + fly,. The energy levels are shown in Figure
412; for a given value of n there are n + 1 approximately
equally spaced levels.
This treatment, like the classical treatment using the coordi
nates £ and r), makes no direct reference to resonance. Let us
1 This example was used by Heisenberg in his first papers on the resonance
phenomenon, Z.f. Phys. 38, 411 (1926); 41. 239 (1927).
XI 41b]
THE RESONANCE PHENOMENON
319
now apply a treatment in which the concept of resonance enters,
retaining the coordinates Xi and x 2 because of their familiar
physical interpretation and applying the methods of approximate
solution of the wave equation given in Chapters VI and VII;
indeed, if the term in X were of more complicated form, it would
be necessary to resort to some approximate treatment. This
term is conveniently considered as the perturbation function in
applying the firstorder perturbation theory. The unperturbed
Ahv c
W
2hi><
hv.
n3
i
<■
n r n n
 1
1
3
 2

— 2
<
Fig. 412. — Energy levels for coupled harmonic oscillators; left, with X » 0;
right, with X « vl/b,
wave equation has as solutions products of Hermite functions
in xi and x 2 ,
<n,(*l, X*) = r ni (xM 2 (*2) =
2 t
— fifL 1 _ aXi
N ni H ni (V*ci)e 2 N nt H ni (\^x 2 )e 2 , (418)
corresponding to the energy values
WTU = (ni + n 2 + l)hp Q = (n + l)hv 0f
with n = tti + n 2 , (419)
the nth level being (n + l)fold degenerate.
The perturbation energy for the nondegenerate level n = is
zero. For the level n = 1 the secular equation is found to be
(Sec. 24)
320 PERTURBATION THEORY INVOLVING THE TIME [XI41b
h\
£ W
= 0,
giving W = ±ftX/2^ . A similar treatment of the succeeding
degenerate levels shows that the firstorder perturbation theory
leads to values for the energy expressed by the first two terms
of Equation 417.
The correct zerothorder wave functions for the two levels
with n = 1 are found to be
and
V2
* A = ^ mxM{x2) ~ ^(ziWfe)},
xps corresponding to the lower of the two levels and \f/ A to the
upper. The subscripts S and A are used to indicate that the
functions are respectively symmetric and antisymmetric in
the coordinates Xi and x 2 . We see that we are not justified in de
scribing the system in either one of these stationary states as con
sisting of the first oscillator in the state n x — 1 and the second in
the state n 2 = 0, or the reverse. Instead, the wave functions
ni = 1, n 2 = and ni = 0, n 2 = 1 contribute equally to each of
the stationary states. It will be shown in Section 41c that if the
perturbation is small we are justified in saying that there is reso
nance between these two states of motion analogous to classical
resonance, one oscillator at a given time oscillating with large
amplitude, corresponding to n x = 1, and at a later time with
small amplitude, corresponding to n\ = 0. The frequency with
which the oscillators interchange their oscillational states, that
is, the frequency of the resonance, is found to be \/v Q} which
is just equal to the separation of the two energy levels divided
by h. This is also the frequency of the classical resonance
(Eq. 415).
In discussing the stationary states of the system of two inter
acting harmonic oscillators we have seen above that it is con
venient to make use of certain wave functions \p n (#i), etc. which
are not correct wave functions for the system, the latter being
XI41b] THE RESONANCE PHENOMENON 321
given by or approximated by linear combinations of the initially
chosen functions, as found by perturbation or variation methods;
and various points of analogy between this treatment and the
classical treatment of the resonating system have been indicated
(see also the following section) . In discussing more complicated
systems it is often convenient to make use of similar methods of
approximate solution of the wave equation, involving the forma
tion of linear combinations of certain initially chosen functions.
The custom has arisen of describing this formation of linear
combinations in certain cases as corresponding to resonance in
the system. In a given stationary state the system is said to
resonate among the states or structures corresponding to those
initially chosen wave functions which contribute to the wave
function for this stationary state, and the difference between
the energy of the stationary state and the energy corresponding
to the initially chosen wave functions is called resonance energy. 1
It is evident that any perturbation treatment for a degenerate
level in which the initial wave functions are not the correct
zerothorder wave functions might be described as involving
the resonance phenomenon. Whether this description would
be applied or not would depend on how important the initial
wave functions seem to the investigator, or how convenient this
description is in his discussion. 2
The resonance phenomenon, restricted in classical mechanics
to interacting harmonic oscillators, is of much greater importance
in quantum mechanics, this being, indeed, one of the most striking
differences between the old and the new mechanics. It arises,
for example, whenever the system under discussion contains two
or more identical particles, such as two electrons or two protons;
and it is also convenient to make use of the terminology in
describing the approximate treatment given the structure of
polyatomic molecules. The significance of the phenomenon for
manyelectron atoms has been seen from the discussion of the
structure of the helium atom given in Chapter VIII; it was there
pointed out (Sec. 29a) that the splitting of levels due to the K
1 There is no close classical analogue of resonance energy.
2 The same arbitrariness enters in the use of the word resonance in describ
ing classical systems, inasmuch as if the interaction of the classical oscillators
is increased the motion ultimately ceases to be even approximately repre
sented by the description of the first paragraph of Section 41a.
322 PERTURBATION THEORY INVOLVING THE TIME [XI41c
integrals was given no satisfactory explanation until the develop
ment of the concept of quantummechanical resonance. The
procedure which we have followed of delaying the discussion
of resonance until after the complete treatment of the helium
atom emphasizes the fact that the resonance phenomenon does
not involve any new postulate or addition to the equations of
wave mechanics but rather only a convenient method of classify
ing and correlating the results of wave mechanics and a basis for
the development of a sound intuitive conception of the theory.
41c. A Further Discussion of Resonance. — It is illuminating
to apply the perturbation method of variation of constants in
order to discuss the behavior of a resonating system. Let us
consider a system for which we have two wave functions, say ^? A
and V Bf corresponding to an energy level of the unperturbed
system with twofold degeneracy. These might, for example,
correspond to the sets of quantum numbers rti — 1, n 2 =0 and
ni = 0, n 2 = 1 for the system of two coupled harmonic oscillators
treated in the previous section. If the perturbation were small,
we could carry out an experiment at the time t = to determine
whether the system is in state A or in state B; for example, we
could determine the energy of the first oscillator with sufficient
accuracy to answer this question. Let us assume that at the
time t = the system is found to be in the state A . We now ask
the following question: On carrying out the investigating experi
ment at a later time t, what is the probability that we would
find that the system is in state A, and what is the probability
that we would find it in state B? In answering this question
we shall see that the physical interpretation of quantummechani
cal resonance is closely similar tp that of classical resonance.
If the perturbation is small, with all the integrals U' mn {m ^ n)
small compared with W Q n — W™ except H' AB and H BA (for which
W° A = Wl), we may assume as an approximation that the
quantities a m (t) remain equal to zero except for a A and a B .
From Equation 396 we see that these two are given by the
equations
2*i,
n l (4110)
6b = — ^(H'^cla + H' AA a B ),}
in which we have taken H BB equal to H AA and H BA equal to H' AB
XI41c] THE RESONANCE PHENOMENON 323
(the system being assumed to consist of two similar parts).
The equations are easily solved by first forming their sum and
difference. The solution which makes cu = 1 and a B = at
t = is
t A Al
a A = e h cos ( — ^t h
a B — —le h sm 
m
(4111)
The probabilities a*a A and a B a B of finding the system in state
A and state B, respectively, at time t are hence
a 2 a A — cosi
\ h '( (4112)
We see that these probabilities vary harmonically between the
values and 1. The period of a cycle (from a*a A = 1 to and
back to 1 again) is seen to be h/2H' AB , and the frequency 2H' AB /h,
this being, as stated in Section 416, just 1/A times the separation
of the levels due to the perturbation.
Let us now discuss in greater detail the sequence of conceptual
experiments and calculations which leads us to the foregoing
interpretation of our equations. Let us assume that we have a
system composed of two coupled harmonic oscillators with
coordinates Xi and x 2y respectively, such that we can at will (by
throwing a switch, say) disengage the coupling, thus causing
the two oscillators to be completely independent. Let us now
assume that for a period of time previous to t = the oscillators
are independent. During this period we carry out a set of two
experiments consisting in separate measurements of the energy
of the oscillators and in this way determine the stationary state
of each oscillator. Suppose that by one such set of experiments
we have found that the first oscillator is in the state n\ = 1
and the second in the state n 2 = 0. The complete system is
then in the physical situation which we have called state A in
the above paragraphs, and so long as the system is left to itself
it will remain in this state.
Now let us switch in the coupling at the time t = 0, and then
switch it out again at the time t = t'. We now, at times later
324 PERTURBATION THEORY INVOLVING THE TIME [XI41c
than t', investigate the system to find what the values of the
quantum numbers n\ and n 2 are. The result of this investigation
will be the same, in a given case, no matter at what time later
than t' the set of experiments is carried out, inasmuch as the
two oscillators will remain in the definite stationary states in
which they were left at time t r so long as the system is left
unperturbed.
This sequence of experiments can be repeated over and over,
each time starting with the system in the state rii — 1, n 2 =
and allowing the coupling to be operative for the length of time t r .
In this way we can find experimentally the probability of finding
the system in the various states n\ = 1, n 2 = 0;rii = 0, n 2 = 1;
rii = 0, n 2 = 0; etc.; after the perturbation has been operative
for the length of time t r .
The same probabilities are given directly by our application of
the method of variation of constants. The probability of
transition to states of considerably different energy as the result
of a small perturbation acting for a short time is very small,
and we have neglected these transitions. Our calculation shows
that the probability of finding the system in the state B depends
on the value of t' in the way given by Equation 4112, varying
harmonically between the limits and 1.
Now in case that we allow the coupling to be operative con
tinuously, the complete system can exist in various stationary
states, which we can distinguish from one another by the measure
ment of the energy of the system. Two of these stationary
states have energy values very close to the energy for the
states rii = 1, n 2 = and n x = 0, n 2 = 1 of the system with
the coupling removed. It is consequently natural for us to
draw on the foregoing argument and to describe the coupled
system in these stationary states as resonating between states
A and B t with the resonance frequency 2H AB /h.
Even when it is not possible to remove the coupling inter
action, it may be convenient to use this description. Thus in
our discussion of the helium atom we found certain stationary
states to be approximately represented by wave functions
formed by linear combination of the wave functions ls(l) 2s(2)
and 2s(l) ls(2). These we identify with states A and B above,
saying that each electron resonates between a Is and a 2s orbit,
the two electrons changing places with the frequency 1/A times
XI41c] THE RESONANCE PHENOMENON 325
the separation of the energy levels ls2s l S and ls2s Z S. It is
obvious that we cannot verify this experimentally, for three
reasons: we cannot remove the coupling, we cannot distinguish
electron 1 from electron 2, and the interaction is so large that
our calculation (based on neglect of all other unperturbed states)
is very far from accurate. These limitations to the physical
verification of resonance must be borne in mind; but they need
not prevent us from making use of the nomenclature whenever
it is convenient (as it often is in the discussion of molecular
structure given in the following chapter).
CHAPTER XII
THE STRUCTURE OF SIMPLE MOLECULES
Of the various applications of wave mechanics to specific
problems which have been made in the decade since its origin,
probably the most satisfying to the chemist are the quantitatively
successful calculations regarding the structure of very simple
molecules. These calculations show that we now have at hand
a theory which can be confidently applied to problems of molec
ular structure. They provide us with a sound conception of the
interactions causing atoms to be held together in a stable mole
cule, enabling us to develop a reliable intuitive picture of the
chemical bond. To a considerable extent the contribution of
wave mechanics to our understanding of the nature of the
chemical bond has consisted in the independent justification of
postulates previously developed from chemical arguments, and
in the removal of their indefinite character. In addition,
wavemechanical arguments have led to the development of many
essentially new ideas regarding the chemical bond, such as the
threeelectron bond, the increase in stability of molecules by
resonance among several electronic structures, and the hybridi
zation of oneelectron orbit als in bond formation. Some of
these topics will be discussed in this chapter and the following
one.
In Sections 42 and 43 we shall describe the accurate and
reliable wavemechanical treatments which have been given the
hydrogen moleculeion and hydrogen molecule. These treat
ments are necessarily rather complicated. In order to throw
further light on the interactions involved in the formation of
these molecules, we shall preface the accurate treatments by a
discussion of various less exact treatments. The helium mole
culeion, He£, will be treated in Section 44, followed in Section 45
by a general discussion of the properties of the oneelectron bond,
the electronpair bond, and the threeelectron bond.
326
XII42a] THE HYDROGEN MOLECULEION 327
42. THE HYDROGEN MOLECULEION
The simplest of all molecules is the hydrogen moleculeion, Hf ,
composed of two hydrogen nuclei and one electron. This mole
cule was one of the stumbling blocks for the old quantum theory,
for, like the helium atom, it permitted the treatment to be carried
through (by Pauli 1 tod Niessen 2 ) to give results in disagreement
with experiment. It was accordingly very satisfying that within
a year after the development of wave mechanics a discussion
of the normal state of the hydrogen moleculeion in complete
agreement with experiment was carried out by Burrau by
numerical integration of the wave equation. This treatment,
together with somewhat more refined treatments due to Hylleraas
A r AB b
Fig. 42—1. — Coordinates used for the hydrogen moleculeion.
and Jaff6, is described in Section 42c. Somewhat simpler and
less accurate methods are described in Sections 42a and 426,
for the sake of the ease with which they can be interpreted.
42a. A Very Simple Discussion. 3 — Following the discussion of
Section 34, the first step in the treatment of the complete wave
equation is the solution of the wave equation for the electron
alone in the field of two stationary nuclei. Using the symbols
of Figure 421, the electronic wave equation is
in which v 2 refers to the three coordinates of the electron and m
is the mass of the electron. 4
1 W. Pauli, Ann. d. Phys. 68, 177 (1922).
2 K. F. Niessen, Dissertation, Utrecht, 1922.
8 L. Pauling, Chem. Rev. 5, 173 (1928).
4 We have included the mutual energy of the two nuclei e*/HAB in this
equation. This is not necessary, inasmuch as the term appears unchanged
in the final expression for W, and the same result would be obtained by
omitting it in this equation and adding it later.
328
THE STRUCTURE OF SIMPLE MOLECULES [XII42a
If Tab is very large, the normal state of the system has the
energy value W = Wn = —Rhc, the corresponding wave
functions being u Ua or ^i v hydrogenatom wave functions
about nucleus A or nucleus B (Sec. 21), or any two independent
linear combinations of these. In other words, for large values
of Tab the system in its normal state is composed of a hydrogen
ion A and a normal hydrogen atom B or of a normal hydrogen
atom A and a hydrogen ion B.
This suggests that as a simple variation treatment of the
system for smaller values of r AB we make use of the same wave
functions Uu a and u u , forming the linear combinations given by
solution of the secular equation as discussed in Section 26<i.
The secular equation is
H AA  W H AB  AW
H BA  AW Ebb  W
= 0,
(422)
in which
and
II a a = jUuHuudr,
Hab = JuujHuujdr,
A = JuuUudr.
A represents the lack of orthogonality of u Ua and Uu . Because
of the equivalence of the two functions, the relations H A a = H B b
and Hab = H B a hold. The solutions of the secular equation are
hence
Haa + Hab
and
Ws
W A =
1+*
Haa — Ha
1  A
These correspond respectively to the wave functions
and
(423)
(424)
(425)
(426)
The subscripts S and ^4 represent the words symmetric and
antisymmetric, respectively (Sec. 29a) ; the wave function p 3 is
XII42a] THE HYDROGEN MOLECULEION 329
symmetric in the positional coordinates of the two nuclei A and B,
and \j/A is antisymmetric in these coordinates.
Introducing Wh by use of the equation
h 2 e 2
8ir'mo A Ta A A
(which is the wave equation for u u ), we obtain for the integral
Haa the expression
Haa = (u u ( Wh  e l + y\ u dr = W H + J + ~ 5 > (427)
in which
In this expression we have introduced in place of r AB the variable
D = r ^ (429)
do
Hba and H A b are similarly given by the expression
Hba = (uuIWh   + —\udr = ATFh + K + ~,
J *\ r B Tab) a CloD
(4210)
in which A is the orthogonality integral, with the value
A = e D (l +D + y 3 D>), (4211)
and K is the integral
K = Juu B (f)u u dT = jf D H + D). (4212)
It is seen that J represents the Coulomb interaction of an
electron in a Is orbital on nucleus A with nucleus B. K may
be called a resonance or exchange integral, since both functions
Uu a and Uu B occur in it.
Introducing these values in Equations 423 and 424, we
obtain
and
w " w ' + £> + T^f • < 42  14 >
330
THE STRUCTURE OF SIMPLE MOLECULES [XII42a
Curves showing these two quantities as functions of r AB are given
in Figure 422. It is seen that yps corresponds to attraction, with
the formation of a stable moleculeion, whereas \f/A corresponds to
repulsion at all distances. There is rough agreement between
observed properties of the hydrogen moleculeion in its normal
state and the values calculated in this simple way. The dis
sociation energy, calculated to be 1.77 v.e., is actually 2.78 v.e.,
and the equilibrium value of r AB , calculated as 1.32 A, is observed
to be 1.06 A.
The nature of the interactions involved in the formation of
this stable molecule (with a one
electron bond) is clarified by the
discussion of a hypothetical case.
Let us assume that our system is
composed of a hydrogen atom
A and a hydrogen ion B, and
that even for small values of r AB
the electron remains attached to
nucleus A, the wave function
being u u . The energy of the
•Q8
0.9
t
W1.0
11
•1.2
A
vN
r o
I 2 5 4 5 6
TAB/dQ— *
Fig. 422. — Energy curves for the
hydrogen moleculeion (in units
eV2ao), calculated for undistorted system Would then be H A a, and
hydrogen atom wave functions. ^ difference be tween this and
W H , namely
H l+ i>
would be the energy of interaction
of a normal hydrogen atom and a hydrogen ion. The curve
representing this energy function, which before the discovery
of the resonance phenomenon was supposed to correspond io
the hydrogen moleculeion, is shown in Figure 422 with the
symbol N. It is seen that it does not correspond to the formation
of a stable bond but instead to repulsion at all distances. The
difference between this curve and the other two is that in this
case we have neglected the resonance of the electron between
the two nuclei A and B. It is this resonance which causes
the actual hydrogen moleculeion to be stable — the energy of the
oneelectron bond is in the main the energy of resonance of the
electron between the two nuclei. (Other interactions, such as
polarization of the atom in the field of the ion, also contribute
to some extent to the stability of the bond. An attempt to
anawer the question of the magnitude of this contribution will
be given in the next section.)
XII42bJ THE HYDROGEN MOLECULEION 331
It is seen from the figure that the resonance interaction sets
in at considerably larger distances than the Coulomb interaction
of atom and ion. This results from the exponential factor
e~ 2D in Haa, as compared with e~ D in the resonance integral K.
For values of r AB larger than 2 A the energy functions Ws and W a
are closely approximated by the values Wh + K and Wh — K,
respectively. In accordance with the argument of Section
416, the resonance energy ±K corresponds to the electron's
jumping back and forth between the nuclei with the frequency
2K/h.
Problem 421. Verify the expressions given for Haa, Hab, and A in
Equations 427 to 4212.
42b. Other Simple Variation Treatments. — We can easily
improve the preceding treatment by introducing an effective
nuclear charge Z'e in the hydrogenlike Is wave functions u u
and u u . This was done by Finkelstein and Horowitz. 1 On
minimizing the energy Ws relative to Z' for various values of
r AB , they obtained a curve for Ws similar to that of Figure 422,
but with a lower minimum displaced somewhat to the left. They
found for the equilibrium value of r AB the value 1.06 A, in com
plete agreement with experiment. The value of the effective
atomic number Z f at this point is 1.228, and the energy of the sys
tem (neglecting oscillational and rotational energy) is — 15.78 v.e.,
as compared with the correct value —16.31 v.e.; the value
of the dissociation energy D e = 2.25 v.e. differing from the cor
rect value 2.78 v.e. by 0.53 v.e. The variation of the effective
atomic number from the value 1 has thus reduced the error
by onehalf.
The energy of the bond for this function too is essentially
resonance energy. Dickinson 2 introduced an additional term,
dependent on two additional parameters, in order to take
polarization into account. He wrote for the (not yet normalized)
variation function
* = u Ua (Z') + uuJLZ') + <t{u 2Pa (Z") + u 2Pb (Z")},
in which the first two terms represent as before Is hydrogenlike
wave functions with effective nuclear charge Z'e and the remain
1 B. N. Finkelstein and G. E. Horowitz, Z.f. Phys. 48, 118 (1928).
* B. N. Dickinson, /. Chem. Phys, 1, 317 (1933).
332 THE STRUCTURE OF SIMPLE MOLECULES [XII42b
u **a =777^(77y""T: r ^ ^os#,
ing two terms functions such as 2p z as described in Section 21,
'Z"\* Z" f''rx
4V27r\ «o / a
in which # is taken relative toaz axis extending from nucleus
A toward nucleus B (and the reverse for u 2Pb ). The parameter
a determines the extent to which these functions enter. The
interpretation of the effect of these functions as representing
polarization of one atom by the other follows from their nature.
The function Uu A + <tu 2Pa differs from u Ua by a positive amount
on the side nearer B and a negative amount on tbe farther side,
in this way being concentrated toward B in the way expected for
polarization. 1
On minimizing the energy relative to the three parameters and
to v% By Dickinson found for the equilibrium distance the value
1.06 A, and for the energy —16.26 v.e., the parameters having
the values Z f = 1.247, Z" = 2.868, and a = 0.145. 2 The
energy calculated for this function differs by only 0.05 v.e.
from the correct value, so that we may say, speaking somewhat
roughly, that the energy of the oneelectron bond is due almost
entirely to resonance of the electron between the two nuclei and
to polarization of the hydrogen atom in the field of the hydrogen
ion, with resonance making the greater contribution (about
2.25 v.e., as given by Finkelstein and Horowitz's function) and
polarization the smaller (about 0.5 v.e.).
It was found by Guillemin and Zener 3 that another variation
function containing only two parameters provides a very good
value for the energy, within 0.01 v.e. of the correct value, the
equilibrium separation of the nuclei being 1.06 A, as for all
functions discussed except the simple one of the preceding
section. This function is
asp ao _L_ /> aog ao
1 The introduction of such a function to take care of polarization was
first made (for the hydrogen molecule) by N. Rosen, Phys. Rev. 38, 2099
(1931).
1 It will be noted that Z" is approximately twice Z'. Dickinson found
that the error in the energy is changed only by 0.02 v.e. by placing Z"
equal to 2Z', the best values of the parameters then being Z' = 1.254,
a = 0.1605.
* V. Guillemin, Jr., and C. Zener, Proc. Nat. Acad. Set. 16, 314 (1929).
XII42c] THE HYDROGEN MOLECULEION 333
the best values of the parameters being Z f = 1.13, Z" = 0.23.
The interpretation of this function is not obvious; we might say,
however, that each of the two terms of the function represents
a polarized hydrogen atom, the first term, for example, being
large only in the neighborhood of nucleus A, and being there
polarized in the direction of nucleus B by the factor e ao
z' r ±
multiplying the hydrogenlike function e ao , the entire wave
function then differing from Dickinson's mainly in the way in
which the polarization is introduced. The value of the principal
effective atomic number Z' = 1.13 is somewhat smaller than
Dickinson's value 1.247.
A still more simple variation function giving better results
has been recently found by James. 1 This function is
<r«(l + of),
in which £ and 77 are the confocal elliptic coordinates defined in
the following section (Eq. 4215), and 8 and c are parameters?
with best values 8 = 1.35 and c = 0.448. The value of the
dissociation energy given by this function is D e — 2,112 v.e.,
the correct value being 2.777 v.e.
42c. The. Separation and Solution of the Wave Equation. — It
was pointed out by Burrau 2 that the wave equation for the
hydrogen moleculeion, Equation 421, is separable in confocal
elliptic coordinates £ and rj and the azimuthal angle <p. The
coordinates £ and 77 are given by the equations
t r A + r B \
f = — T 7
AB } (4215)
Ta — r B I
*? = — r )
Tab I
On introduction of these coordinates (for which the Laplace
operator is given in Appendix IV), the wave equation becomes
¥%&{%&  ,■> + £«}*  0, (4216)
1 H. M. James, private communication to the authors.
1 0yvind Burrau, Det. Kgl. Danske Vid. Selskab. 7, 1 (1927).
334 THE STRUCTURE OF SIMPLE MOLECULES [XII42c
in which we have made use of the relation
f! +^! = 4e 2 ^
Ta r B r AB (Z 2  v 2 )
and have multiplied through by • , — — • The quantity
W f , given by
pp = W  — , (4217)
Tab
is the energy of the electron in the field of the two nuclei, the
mutual energy of the two nuclei being added to this to give the
total energy W.
It is seen that on replacing i^(£, ?j, <p) by the product function
*({, u, *>) = E({)H(i,)*(?) (4218)
this equation is separable 1 into the three differential equations
B =  m2 *' (42_19)
{(l^} + (x^ r ^.)H=0, (4220)
{« 2 " d } + (~^ + *>* ~ w^j + ") s = °' (42 " 21)
and
in which
and
X = ,^ (4222)
D = ^2. (4223)
a
The range of the variable J is from 1 to °o , and of rj from — 1 to
+ 1. The surfaces £ = constant are confocal ellipsoids of revolu
tion, with the nuclei at the foci, and the surfaces t\ = constant
are confocal hyperboloids. The parameters m, X, and /z must
assume characteristic values in order that the equations possess
acceptable solutions. The familiar <p equation possesses such
solutions for m = 0, ±1, ±2, • • • . The subsequent procedure
of solution consists in finding the relation which must exist
1 The equation is also separable for the case that the two nuclei have
different charges.
XII42cl
THE HYDROGEN MOLECULEION
335
08
between X and /x in order that the t\ equation possess a satis
factory solution, and, using this relation, in then finding from the
f equation the characteristic values of X and hence of the energy.
This procedure was carried out for the normal state of the
hydrogen moleculeion by Burrau in 1927 by numerical integra
tion of the £ and rj equations. More accurate treatments have
since been given by Hylleraas 1 and by Jaflte. 2 (The simple
treatment of Guillemin and Zener, described in the preceding
fection, approaches Burrau's
in accuracy.) We shall not
describe these treatments in
detail but shall give a brief dis
cussion of one of them (that
of Hylleraas) after first pre
senting the results.
The energy values calculated
by the three authors are given
in Table 421 and shown graph
ically in Figure 423. It is seen
that the curve is qualitatively
similar to that given by the very
0.9
wt
1.0
1.2
1.3
1
r AB/a
ix A , ro x. a* FlG ' 423.— The energy of the
Simple treatment Ot bection 42a normal hydrogen moleculeion (in units
(Fig. 422). The three treat e2/2ao > as a function of tab.
ments agree in giving for the equilibrium value 3 of r AB 2.00 a or
1.06 A, as was found for the variation functions of the preceding
section also. This is in complete agreement with the band
spectral value. Spectroscopic data have not been obtained for
the hydrogen moleculeion itself but rather for various excited
states of the hydrogen molecule. It is believed that these are
states involving a normal hydrogen moleculeion as core, with a
highly excited outer electron in a large orbit, having little effect
1 E. A. Hylleraas, Z. f. Phys. 71, 739 (1931).
2 G. Jaffe, Z. f. Phys. 87, 535 (1934).
3 The average value of tab for various oscillational states as determined
from bandspectral data is found to be a function of the vibrational quantum
number v, usually increasing somewhat with increasing v. The value
for v = is represented by the symbol r , and the extrapolated value corre
sponding to the minimum of the electronic energy function by the symbol
r 6 . The vibrational frequencies are similarly represented by *> and v 4
(or by wo and « a , which have found favor with band spectroscopists) and
the energies of dissociation by Do and D«.
336
THE STRUCTURE OF SIMPLE MOLECULES [XII 42c
on the potential function for the nuclei; this belief being supported
by the constancy of the values of r e and v e (the oscillational
frequency) shown by them. The values of r e were extrapolated
by Birge 1 and Richardson 2 to give 1.06 A for the moleculeion.
Table 421. — Electronic Energy Values for the Hydrogen
Moleculeion
Wiij (in units Rhc)
tab/cio
Burrau
Guillemin
and Zener
Hylleraas
Jafte
0.5
00
0.896
1 195*
1 204
1.198*
1.000
00
00
0.5302
0.9046
1.0826
1.1644
1.1980
 1 . 20527
 1 . 1998
1.1878
1.1716
1.1551
1.0000
00
5300
1.0
1.25
0.903
0.9035
1.5
1.75
2.00
2.25
2 5
1.198f
1.205
— 1 . 197f
 1 . 20528
2.75
3.0
00
1 154
1 000
1.1544
1 0000
* Interpolated between adjacent values calculated by Burrau, who estimated his accu
racy in the neighborhood of the minimum as ± 0.002 Rhc.
t Interpolated values.
The value — 1.20528jR/ic for the energy of the moleculeion is
also substantiated by experiment; the discussion of this com
parison is closely connected with that for the hydrogen molecule,
and we shall postpone it to Section 43d. The behavior of the
minimum, however, can be compared with experiment by way
of the vibrational energy levels. By matching a Morse curve
to his calculated points and applying Morse's theory (Sec. 35d),
Hylleraas found for the energy of the molecule ion in successive
vibrational levels given by the quantum number v the expression
W v = 1.20527 + 0.0206(y + Y 2 )  0.000510 + y 2 )\ (4224)
in units Rhc. This agrees excellently with the expressions
obtained by Birge 1 and Richardson 2 by extrapolation of the
observed vibrational levels for excited states of the hydrogen
1 R. T. Birge, Proc. Nat. Acad. Set. 14, 12 (1928).
*0. W. Richardson, Trans. Faraday Soc. 26, 686 (1929).
XII42C
THE HYDROGEN MOLECULEION
337
molecule, their coefficients in these units being 0.0208 and
0.00056, and 0.0210 and 0.00055, respectively.
The value W e = — 1. 20527 R H hc corresponds to an electronic
energy of the normal hydrogen moleculeion of —16.3073 v.e.
(using Rnhc = 13.5300 v.e.) and an electronic dissociation energy
into H + H+ of D e = 2.7773 v.e., this value being shown to be
accurate to 0.0001 v.e. by the agreement between the calculations
of Hylleraas and Jaff6. The value of Z> , the dissociation energy
of the moleculeion in its lowest vibrational state, differs from
this by the correction terms given in Equation 4224. These
Fig. 424. — The electron distribution function for the normal hydrogen
moleculeion (Burrati). The upper curve shows the value of the function along
the line passing through the two nuclei, and the lower figure shows contour
lines for values 0.9, 0.8, • • • ,0.1 times the maximum value.
terms are not known so accurately, either theoretically or
experimentally. Hylleraas's values lead to a correction of
0.138 v.e., Birge's to 0.139 v.e., and Richardson's to 0.140 v.e.
If we accept the theoretical value 0.138 ± 0.002 v.e. we obtain
Do = 2.639 ± 0.002 v.e.
as the value of the dissociation energy of the normal hydrogen
moleculeion.
The wave function for the normal moleculeion as evaluated by
Burrau corresponds to the electron distribution function repre
sented by Figure 424. It is seen that the distribution is
closely concentrated about the line between the two nuclei,
the electron remaining most of the time in this region.
Let us now return to a brief discussion of one of the accurate
treatments of this system, that of Hylleraas, which illustrates
338 THE STRUCTURE OF SIMPLE MOLECULES [XII42c
the method of approximate solution of the wave equation dis
cussed in Section 27a.
The variable r; extends through the range —1 to +1, which
is the range traversed by the argument z = cos & of the associated
Legendre functions Pi mI of Section 19. With Hylleraas we
expand the function H(r/) in terms of these functions, writing
00
HW = X c i p \ m ^), (4225)
iIH
in which the coefficients c t are constants. Substituting this
expression in Equation 4220, and simplifying with the aid of the
differential equation satisfied by the associated Legendre func
tions, Equation 199, we obtain the equation
00
2 c,{W nl(]i + IJIWn) = 0. (4226)
im
We can eliminate the factor j; 2 by the use of the recursion formula
(I  \m\ + l)(l  \m\ + 2) ,
(21 + 1)(2I + 3)
„»PMM = St ~ m ^ ill!. ~ l w ' t *> p\»>i
( q _ \ m \ + m + H + 1) (I  \m\)(l + mf)  p,„
+ \ (2 + l)(H + 3) + (21  1)(2J + 1) f '
I (l + \m\)(l + \m\l)
+ (21  l)(2l + 1) ^" 2J (42 ^ 27)
which is easily obtained by successive application of the ordinary
recursion formula 1916. On introducing this in Equation
4226, it becomes a simple series in the functions Pi ml M with
coefficients independent of r\. Because of the orthogonality of
these functions, their coefficients must vanish independently in
order that the sum vanish (Sec. 22). This gives the condition
(l\m\l)(l\m\\ „ , r/ (j ~ H + W + H + 1)
(21  3) (a  i) l ' 2 + [\ (a + i)(a + 3)
which is a threeterm recursion formula in the coefficients Ci.
XH42c]
THE HYDROGEN MOLECULEION
339
We now consider the set of equations 4228 for different values
of I as a set of simultaneous linear homogeneous equations in the
unknown quantities ci. In order that the set may possess a non
trivial solution, the determinant formed by the coefficients of
the Ci'b must vanish. This gives a determinantal equation
involving X and n, from which we determine the relation between
them.
We are interested in the normal state of the system, with m =
and I even. The determinantal equation for this case is
X — n
11
21
15
X  6  m
35 X
39
77
X  20  /x
= 0.
(4229)
The only nonvanishing terms are in the principal diagonal and
the immediately adjacent diagonals. As a rough approximation
(to the first degree in X) we can neglect the adjacent diagonals;
the roots of the equation are then \i = J^X, m = x Ki^ — 6,
M = 3 ^77^ ~~ 20, etc. We are interested in the first of these.
In order to obtain it more accurately, we solve the equation again,
including the first two nondiagonal terms, and replacing \x in
the second diagonal term by J^X. This equation,
= 0,
M = MX + K35X 2 + HsosX*,
in which powers of X higher than the third are neglected. Hyller
aas carried the procedure one step farther, obtaining
M = y z \ + % 35 x 2 + ^ 5 05* 3  0.000013X 4  0.0000028X 5 .
This equation expresses the functional dependency of m on X
for the normal state, as determined by the 77 equation. The
next step is to introduce this in the £ equation, eliminating /x,
and then to solve this equation to obtain the characteristic
3XM
£
1*
21 X
6ix
has the solution
340 THE STRUCTURE OF SIMPLE MOLECULES [XII43a
values of X and hence of the energy as a function of r AB . Because
of their more difficult character, we shall not discuss the methods
of solution of this equation given by Hylleraas and Jafife.
42d. Excited States of the Hydrogen Moleculeion. — We
have discussed (Sec. 42a) one of the excited electronic states of
the hydrogen moleculeion, with a nuclearantisymmetric wave
function formed from normal hydrogenatom functions. This
is not a stable state of the moleculeion, inasmuch as the potential
function for the nuclei does not show a minimum.
Calculations of potential functions for other excited states,
many of which correspond to stable states of the moleculeion,
have been made by various investigators, 1 among whom
Teller, Hylleraas, and JafT6 deserve especial mention.
43. THE HYDROGEN MOLECULE
43a. The Treatment of Heitler and London. — The following
simple treatment of the hydrogen molecule (closely similar to
that of the hydrogen moleculeion described in Section 42a)
does not differ essentially from that given by Heitler and London 2
in 1927, which marked the inception (except for Burrau's earlier
paper on the moleculeion) of the application of wave mechanics
to problems of molecular structure and valence theory. Heitler
and London's work must be considered as the greatest single contri
bution to the clarification of the chemist's conception of valence
which has been made since G. N. Lewis's suggestion in 1916 that
the chemical bond between two atoms consists of a pair of
electrons held jointly by the two atoms.
Let us first consider our problem with neglect of the spin of
the electrons, which we shall then discuss toward the end of the
section. The system comprises two hydrogen nuclei, A and B,
and two electrons, whose coordinates we shall designate by the
symbols 1 and 2. Using the nomenclature of Figure 431, the
wave equation for the two electrons corresponding to fixed posi
tions of the two nuclei is
1 P. M. Morse and E. C. G. Stueckelberg, Phys. Rev. 33, 932 (1929);
E. A. Hylleraas, Z.J. Phys. 61, 150 (1928); 71, 739 (1931); J. E. Lennard
Jones, Trans. Faraday Soc. 24, 668 (1929); E. Teller, Z. J. Phys. 61, 458
(1930); G. Jaffe, Z.J. Phys. 87, 535 (1934).
1 W. Heitler and F. London, Z. J. Phys. 44, 455 (1927).
XII 43a]
THE HYDROGEN MOLECULE
341
V?* + V?* +
87r 2 m J
Tax r B \ r A i r B 2 f*i 2
£}<=»•
(431)
For very large values of r AB we know that in its normal state the
system consists of two normal hydrogen atoms. Its wave func
tions (the state having twofold degeneracy) are then u u (1)
Uu b (2) and u Ub (1) u Ua (2) or any two independent linear com
binations of these two (the wave function u Ua (1) representing a
hydrogenlike Is wave function for electron 1 about nucleus A,
A r AB B
Fig. 431. — Coordinates used for the hydrogen molecule, represented diagram
matically.
etc., as given in Section 21). This suggests that for smaller
values of t A b we use as variation function a linear combination of
these two product functions. We find as the secular equation
corresponding to this linear variation function (Sec. 26c?)
HnW #i n  A 2 Tf
= 0,
(432)
in which
and
with
Hi i = ffxpiHfadndrz,
Hi ii = Jj\l/iH\l/ndTidT 2j
A 2 = JfhhidTidT2,
h = Uu A (l)uu a (2) and ^ n = 1^(1)^(2).
It is seen that A is the orthogonality integral introduced in Section
42a, and given by Equation 4211. With Hi i = Hu n and
Hi ii = Hn i, the equation can be immediately solved to give
342 THE STRUCTURE OF SIMPLE MOLECULES [XII48a
and
w _ Hi i + Hi ii ( , Q „v
Ws 1 + A » ( 3 ~ 3 '
TI7 Hi l ~~ Hi U /AO A\
Wa = — i  a 2 ^ (434)
corresponding to the wave functions
*, = —L^ [uu A (l)u l§B (2) + u Ub (1)uu a (2) } (435)
and
+ A = ^={^(1)^(2)  u Ub (1)uu a (2)}. (436)
^s is symmetric in the positional coordinates of the two electrons
and also in the positional coordinates of the two nuclei, whereas
\(/a is antisymmetric in both of these sets of coordinates.
On evaluation we find for Hi i the expression
ft i = f f «i..(D«,.,(2)(W„ **+*.+ *)
J J \ r B \ r A 2 r 12 Tab/
Wi^(l)wi. fl (2)dr 1 dr 2
= 2W H + 2J + J' + — , (437)
Tab
in which J is the integral of Equation 428 and J' is
="5{b«Kb + T + I c +H}' (43  8)
with D as before equal to TabAo.
Similarly we find for Hi n the expression
ft„ = JJ^(l) M ,,(2)(2^£^ + £
— JWi^(l)u 1 . A (2)dridr 1
 2A 2 TF* + 2A# + £' + A 2 —, (439)
in which if is the integral of Equation 4212 and K' is
Xn48a] THE HYDROGEN MOLECULE 343
,(1^(2)11^(1^(2)
£[«■< ¥ + f o+3D,+ H
+ ~{A 8 (7 + log D) + A'*Ei(4D)  2AA , ^(~2D)} 1,
(4310)
in which y =0.5772 • • • is Euler's constant and
A'  e D (l  D + HD*).
Ei is the function known as the integral logarithm. 1 (The
integral K' was first evaluated by Sugiura, 2 after Heitler and
London had developed an approximate expression for it.) J'
represents the Coulomb interaction of an electron in a Is orbital
on nucleus A with an electron in a Is orbital on nucleus
By and K f is the corresponding resonance or exchange integral.
Substituting these values in Equations 433 and 434, we
obtain
Ws  2W " + 7T B + «x7 T (43iD
and
e
2J + J'
+ 2AK + K'
1
+ A 2
2J + J'
 2AK
K'
Wa = 2Wh + fz> + i ~ — • ( 43  12 >
Curves representing Ws and Wa as functions of r AB are shown in
Figure 432. It is seen that Wa corresponds to repulsion at all
distances, there being no equilibrium position of the nuclei.
The curve for Wa corresponds to attraction of the two hydrogen
atoms with the formation of a stable molecule, the equilibrium
value calculated for t A b being 0.80 A, in rough agreement with
the experimental value 0.740 A, The energy of dissociation of
the molecule into atoms (neglecting the vibrational energy of the
nuclei) is calculated to be 3.14 v.e., a value somewhat smaller
than the correct value 4.72 v.e. The curvature of the potential
function near its minimum corresponds to a vibrational frequency
for the nuclei of 4800 cm 1 , the bandspectral value being
4317.9 cm 1 .
It is seen that even this very simple treatment of the problem
leads to results in approximate agreement with experiment.
1 See, for example, Jahnke and Emde, << Funktionentafeln. , '
* Y. Suqiura, Z. /. Phy8. 46, 484 (1927).
344
THE STRUCTURE OF SIMPLE MOLECULES [XII43a
It may be mentioned that the accuracy of the energy calculation
is greater than appears from the values quoted for D ej inasmuch
as the energy of the electrons in the field of the two nuclei
(differing from Wa by the term e 2 /r AB ) at r AB = 1.5a is calculated
to be 2W H — 18.1 v.e., and the error of 1.5 v.e. is thus only a
few per cent of the total electronic interaction energy.
It is interesting and clarifying for this system also (as for
the hydrogen moleculeion) to discuss the energy function for a
hypothetical case. {Let us suppose that the wave function for
the system were fa = ^i^(l) u Ub (2) alone. The energy of the
system would then be Hi i, which is shown as curve N in Figure
432. (it is seen that this curve
1.4
1.6
w
2.0
2.2
24
1
yA
i"
v"
I
gives only a small attraction
between the two atoms, with a
bond energy at equilibrium only
a few per cent of the observed
value. The wave function fa
differs from this function in the
interchange of the coordinates
of the electrons, and we conse
quently say that the energy
r AB/a ~+* of the bond in the hydrogen
Fio. 432.— Energy curves for the molecule is in the main reso
hydrogen molecule (in units e 2 /2ao). ~ * . « i
nance or interchange energy.
So far we have not taken into consideration the spins of the
electrons. On doing this we find, exactly as for the helium
atom, that in order to make the complete wave functions anti
symmetric in the electrons, as required by Pauli's principle, the
orbital wave functions must be multiplied by suitably chosen
spin functions, becoming
1
and
fa~{a(l)P(2) j3(l)a(2)}
V2
^•«(l)a(2),
^~{a(l)«2)+j9(l)a(2)},
1^.0(1)0(2).
There are hence three repulsive states A for one attractive
state S; the chance is % that two normal hydrogen atoms
XII43b]
THE HYDROGEN MOLECULE
345
will interact with one another in the way corresponding to the
formation of a stable molecule. It is seen that the normal state
of the hydrogen molecule is a singlet state, the spins of the
two electrons being opposed, whereas the repulsive state A is p
triplet state.
43b, Other Simple Variation Treatments. — The simple step
of introducing an effective nuclear charge Z'e in the Is hydrogen
like wave functions of 435 was taken by Wang, 1 who found that
this improved the energy somewhat, giving D e = 3.76 v.e.,
and that it brought the equilibrium internuclear distance r e
down to 0.76 A, only slightly
greater than the experimental
value 0.740 A. The effective
nuclear charge at the equilib
rium distance was found to be
Z e = 1.1666.
There exists the possibility
that wave functions correspond
ing to the ionic structures H~H +
and H + H~ might also make
5ve
10
15
1
5A
r AB"*^
Fig. 433.— The mutual Coulomb
an appreciable Contribution to energy of two ions with charges He
ji r x • r ii and — e as a function of txb>
the wave function tor the nor
mal state of the molecule. These ionic functions arewi 8 (1) u u (2)
and Uu B (l) Uu B (2), the corresponding spin function allowed
by Pauli's principle being — { a (l) 0(2)  0(1) a(2)}, as for
V2
fa. It is true that for large values of t A b the energy of the ionic
functions is 12.82 v.e. greater than that for the atomic functions,
this being the difference of the ionization potential and the
electron affinity (Sec. 29c) of hydrogen; but, as r AB is decreased,
the Coulomb interaction of the two ions causes the energy for the
ionic functions to decrease rapidly, as shown in Figure 433,
the difference of 12.82 v.e. being counteracted at 1.12 A. This
rough calculation suggests that the bond in the hydrogen molecule
may have considerable ionic character, the structures H~H + and
H+H~ of course contributing equally. The wave function
Uu A (l) u Ub (2) + Uu B (l) u Ua (2) + c{u,^(l) Uu A (2) +
Uu B (l) Uu B (2)} (4313)
1 S. C. Wang, Phys. Rev. 31, 579 (1928).
346
THE STRUCTURE OF SIMPLE MOLECULES [XII43b
2.16
was considered by Weinbaum, 1 using an effective nuclear charge
Z'e in all the Is hydrogenlike functions. On varying the param
eters, he found the minimum of the energy curve (shown in
Figure 434) to lie at t A b — 0.77 A, and to correspond to the
value 4.00 v.e. for the dissociation energy D e of the molecule.
This is an appreciable improvement, of 0.24 v.e., over the value
given by Wang's function. The parameter values minimizing
the energy 2 were found to be
Z' = 1.193 and c = 0.256.
It may be of interest to
consider the hydrogenmole
cule problem from another
point of view. So far we have
attempted to build a wave func
tion for the molecule from
atomic orbital functions, a pro
cedure which is justified as a
first approximation when r A B
is large. This procedure, as
generalized to complex mole
cules, is called the method of
2.20
w.
■2.24
2.28
2.32
A
1.00
L75
1.25 1.50
r A67a —
Fig. 434. — Energy curves for the
hydrogen molecule (in units e 2 /2ao) : A,
for an extreme molecularorbital wave
function; B, for an extreme valence
bond wave function; and c, for a valencebond wave functions, the
valencebond function with partial ionic _ rt _^ ™~^+,_^ U^,*~~ „„~A i~
character ( Weinbaum) . name sometimes being used in
the restricted sense of implying
neglect of the ionic terms. Another way of considering the
structure of complex molecules, called the method of molecular
orbitals,* can be applied to the hydrogen molecule in the following
way. Let us consider that for small values of K r AB the interaction
of the two electrons with each other is small compared with their
interaction with the two nuclei. Neglecting the term e 2 /r 12
in the potential energy, the wave equation separates into equa
tions for each electron in the field of the two nuclei, as in the
hydrogenmoleculeion problem, and the unperturbed wave
function for the normal state of the molecule is seen to be the
1 S. Weinbaum, J. Chem. Phys. 1, 593 (1933).
* Weinbaum also considered a more general function with different
effective nuclear charges for the atomic and the ionic terms and found that
this reduced to 4313 on variation.
« F. Hund, Z. f. Phys. 51, 759 (1928); 73, 1 (1931); etc.; R. S. Mulliken,
Phys. Rev. 32, 186, 761 (1928); 41, 49 (1932); etc.; M. Dunkel, Z.f. phys.
Chem. B7, 81; 10, 434 (1930); E. Huckel, Z.f. Phys. 60, 423 (1930); etc.
XII43b] THE HYDROGEN MOLECULE 347
product of normal hydrogenmoleculeion wave functions for the
two electrons. Inasmuch as the function u u (1) + ^i« (1)
is a good approximation to the wave function for the electron
in the normal hydrogen moleculeion, the molecularorbital
treatment corresponds to the wave function
\uu A 0) + Uu M Q) i {uu A &) + u Ub {2) } (4314)
for the normal hydrogen molecule. It is seen that this is identical
with Weinbaum's function 4313 with c = 1; that is, with the
ionic terms as important as the atomic terms.
If the electric charges of the nuclei were very large, the inter
electronic interaction term would actually be a small perturba
tion, and the molecularorbital wave function 4314 would be a
good approximation to the wave function for the normal state
of the system. In the hydrogen molecule, however, the nuclear
charges are no larger than the electronic charges, and the mutual
repulsion of the two electrons may well be expected to tend to
keep the electrons in the neighborhood of different nuclei, as
in the simple HeitlerLondonWang treatment. It would be
difficult to predict which of the two simple treatments is the
better. On carrying out the calculations 1 for the molecular
orbital function 4314, introducing an effective atomic number
Z' y the potential curve A of Figure 434 is obtained, correspond
ing to r e = 0.73 A, D e = 3.47 v.e., and Z' = 1.193. It is seen
that the extreme atomicorbital treatment (the Wang curve) is
considerably superior to the molecularorbital treatment for the
hydrogen molecule. 2 This is also shown by the results for the
more general function 4313 including ionic terms with a coeffi
cient c; the value of c which minimizes the energy is 0.256,
which is closer to the atomicorbital extreme (c = 0) than to the
molecularorbital extreme (c = 1).
For the doubly charged helium moleculeion, He^~+, a treatment
Dased on the function 4313 has been carried through, 3 leading
to the energy curve shown in Figure 435. It is seen that at
large distances the two normal He+ ions repel each other with the
force e 2 /r 2 . At about 1.3 A the effect of the resonance integrals
1 For this treatment we are indebted to Dr. S. Weinbaum.
2 Similar conclusions are reached also when Z' is restricted to the value
1 (HeitlerLondon treatment).
*L. Pauling, /. Chem. Phys. 1, 56 (1933).
348
THE STRUCTURE OF SIMPLE MOLECULES [XII43b
becomes appreciable, leading to attraction of the two ions and a
minimum in the energy curve at the predicted internuclear
equilibrium distance r e = 0.75 A (which is very close to the value
for the normal hydrogen molecule). At this distance the values
of the parameters which minimize the energy are Z' = 2.124
and c = 0.435. This increase in the value of c over that for the
hydrogen molecule shows that as a result of the larger nuclear
charges the ionic terms become more important than for the
hydrogen molecule.
Fig. 435. — The energy curve for normal He^+.
We have discussed the extension of the extreme atomicorbital
treatment by the inclusion of ionic terms. A further extension
could be made by adding terms corresponding to excited states
of the hydrogen atoms. Similarly the molecularorbital treat
ment could be extended by adding terms corresponding to
excited states of the hydrogen moleculeion. With these
extensions the treatments ultimately become identical. 1 In
the applications to complex molecules, however, it is usually
practicable to carry through only the extremely simple atomic
orbital and molecularorbital treatments; whether the slight
superiority indicated by the above considerations for the atomic
1 See J. C. Slater, Phys. Rev. 41, 255 (1932).
Xn43c]
THE HYDROGEN MOLECULE
349
orbital treatment is retained also for molecules containing
atoms of larger atomic number remains an open question.
So far we have not considered polarization of one atom by the
other in setting up the variation function. An interesting
attempt to do this was made by Rosen, 1 by replacing u u (1) in
the HeitlerLondonWang function by u Uj (l) + (tu 2Pa (1) (with
a similar change in the other functions), as in Dickinson's
treatment of the hydrogen moleculeion (which was suggested
by Rosen's work). The effective nuclear charges Z'e in u u
and Z"e in u 2p were assumed to be related, with Z" = 2Z'.
It was found that this leads to an improvement of 0.26 v.e. in
the value of D e over Wang's treatment, the minimum in the
energy corresponding to the values r e = 0.77 A, D e = 4.02 v.e.,
Z' = 1.19, and a = 0.10.
A still more general function, obtained by adding ionic terms
(as in 4313) to the Rosen function, was discussed by Weinbaum,
who obtained D e = 4.10 v.e., Z' = 1.190, a = 0.07, and c = 0.176.
The results of the various calculations described in this section
are collected in Table 431, together with the final results of
James and Coolidge (see following section).
Table 431. — Results of Approximate Treatments of the Normal
Hydrogen Molecule
HeitlerLondonSugiura
Molecularorbital treatment . .
Wang
Weinbaum (ionic)
Rosen (polarization)
Weinbaum (ionicpolarization)
JamesCoo lidge
Experiment
D e
3. 14 v.e.
3.47
3.76
4.00
4.02
4.10
4.722
4.72
0.80 A
0.73
0.76
0.77
0.77
0.74
0.7395
4800 era"
4900
4750
4260
4317.9
1.193
1.166
1.193
1.19
1.190
43c. The Treatment of James and Coolidge. — In none of the
variation functions discussed in the preceding section does the
intereleetronic interaction find suitable expression. A major
advance in the treatment of the hydrogen molecule was made
by James and Coolidge 2 by the explicit introduction of the
1 N. Rosen, Phys. Rev. 38, 2099 (1931).
a H. M. James and A. S. Coolidge, /. Chem. Phys. 1, 825 (1933).
350 THE STRUCTURE OF SIMPLE MOLECULES [XII43c
interelectronic separation r i2 in the variation function (the
similar step by Hylleraas having led to the ultimate solution
of the problem of the normal helium atom). Using the elliptic
coordinates (Sec. 42c)
t r AX + r Bl r A2 + r B2
Tab Tab
Ta\ — r B \ Tai — r B 2
rji = ; rj 2 = >
Tab Tab
and the new coordinate
„ 2ri2
U = y
Tab
James and Coolidge chose as the variation function the expression
* = ^~ a(fl+f,) 2 Cmn; ^ (ff ^ r?Jl?7 ^ P + ^^"^ p ), (4315)
mnjkp
the summation to include zero and positive values of the indices,
with the restriction that j + k be even, which is required to make
the function symmetric in the coordinates of the nuclei.
Calculations were first made for r AB = 1.40a (the experi
mental value of r e ) and 8 = 0.75; with these fixed values the
variation of the parameters can be carried out by the solution
of a determinantal equation (Sec. 26d). It was found that
five terms alone lead to an energy value much better than any
that had been previously obtained, 1 the improvement being due
in the main to the inclusion of one term involving u (Tables
432 and 433). It is seen from Table 432 that the eleventerm
and thirteenterm functions lead to only slightly different energy
values, and the authors' estimate that the further terms will
contribute only a small amount, making D e = 4.722 ± 0.013 v.e.,
seems not unreasonable.
Using the eleventerm function, James and Coolidge investi
gated the effects of varying 5 and r AB , concluding that the
values previously assumed minimize the energy, corresponding
to agreement between the theoretical and the experimental
value of r c , and that the energy depends on tab in such a way as to
correspond closely to the experimental value of v e .
1 It is of interest that the best value found by including only terms with
p = is D e  4.27 v.e., which is only slightly better than the best values
of the preceding section.
XII 43d]
THE HYDROGEN MOLECULE
351
This must be considered as a thoroughly satisfactory treat
ment of the normal hydrogen molecule, the only improvement
which we may look forward to being the increase in accuracy by
the inclusion of further terms.
Table 432. — Successive Approximations with the JamesCoolidge
Wave Function for the Hydrogen Molecule
Number of terms
Total energy
D.
1
5
11
13
2.189 R H hc
2.33290
2.34609
2.34705
2.56 v.e.
4.504
4.685
4.698
Table 433. — Values of Coefficients c mn jkp for Normalized Wave
Functions for the Hydrogen Molecule*
Term mnjkp
00000
00020
10000
00110
00001
10200
10110
10020
20000
00021
10001
00002
00111
Values of c mn jk P
1 term
1.69609
5 terms
2.23779
0.80483
0.60985
0.27997
0.19917
11 terms
2.29326
1 . 19526
0.86693
0.49921
0.33977
0.13656
0.14330
0.07214
0.06621
0.02456
0.03143
13 terms
2.22350
1 . 19279
0.82767
0.45805
0.35076
0.17134
0.12394
0.12101
0.08323
0.07090
0.03987
0.01197
0.01143
*In a later note, J. Chem. Phys. 3, 129 (1935), James and Coolidge state that these
values are about 0.05 per cent too large.
43d. Comparison with Experiment. — The theoretical values
for the energy of dissociation of the hydrogen molecule and mole
culeion discussed in the preceding sections can be compared with
experiment both directly and indirectly. The value
Do = 2.639 ± 0.002 v.e.
for H£ is in agreement with the approximate value 2.6 ±0.1 v.e.
found from the extrapolated vibrational frequencies for excited
352
THE STRUCTURE OF SIMPLE MOLECULES [XII43d
states of H 2 . For the hydrogen molecule the energy calculations
of James and Coolidge with an estimate of the effect of further
terms and corrections for zeropoint vibration (using a Morse
function) and for the rapid motion of the nuclei (corresponding
to the introduction of a reduced mass of electron and proton)
lead to the value 1 4.454 ± 0.013 v.e. for the dissociation energy
O . This is in entire agreement with the most accurate experi
Ht + E"
H+H + +E"
I
v /
i
D e (H
v=o
2 + )
)
Do(H e
e(H 2 + )
i
i '
A
4hi
1(H)
KH 2 )
H+H
1
i
D e (H 2 )
[
w
)
'
u
H»
' ,'t
^hT) e (H 2 )
Fig. 436. — Energylevel diagram for a system of two electrons and two protons.
mental value, 4.454 ± 0.005 v.e., obtained by Beutler 2 by the
extrapolation of observed vibrational levels.
Another test of the values can be made in the following way.
From the energylevel diagram for a system of two electrons and
1 H. M. James and A. S. Coolidge, J. Chem. Phys. 3, 129 (1935). We are
indebted to Drs. James and Coolidge for the personal communication of this
and other results of their work.
2 H. Beutler, Z. phys. Chem. B27, 287 (1934). A direct thermochemical
determination by F. R. Bichowsky and L. C. Copeland, Jour. Am. Chem.
Soc. 50. 1315 (1928), gave the value 4.55 ± 0.15 v.e.
XII43e] THE HYDROGEN MOLECULE 353
two protons shown in Figure 436 we see that the relation
J(H 2 ) + D (H+) = /(H) + D (H 2 ) (4316)
holds between the various ionization energies and dissociation
energies. With the use of the known values of 7(H) and D (HJ)
(the latter being the theoretical value) and of the extrapolated
spectroscopic value of 7(H 2 ), £>o(H 2 ) is determined 1 as
4.448 ± 0.005 v.e.,
again in excellent agreement with the value given by James
and Coolidge.
43e. Excited States of the Hydrogen Molecule. — Several
excited states of the hydrogen molecule have been treated by
perturbation and variation methods, 2 with results in approximate
agreement with experiment.
Instead of discussing these results, let us consider the simple
question as to what wave functions for the hydrogen molecule
can be built from Is hydrogenlike functions u, L and u B alone.
There are four product functions of this type, u A (l)u B (2),
u B (l)u A (2), u A (l)u A (2), and u B (\)u B (2). The equivalence of
the two electrons and of the two nuclei requires that the wave
functions obtained from these by solution of the secular equation
be either symmetric or antisymmetric in the positional coordi
nates of the two electrons and also either symmetric or antisym
metric in the two nuclei. These functions are
n}'
^(1)^(2) + u B (l)u A (2)\, {u a (\)u a (2) +^(1)^(2))
\S N S E1 2+,
III ^(1)^(2)  u B (l)u A (2),A N A E *Xt,
IV u A (l)u A (2)  u B (l)u B (2),A»S* l 2i 9
functions I and II being formed by linear combination of the two
indicated functions. One of these (I, say) represents the
1 Personal communication to Dr. James from Prof. O. W. Richardson.
2 E. C. Kemble and C. Zener, Phys. Rev. 33, 512 (1929); C. Zener and
V\ Guillemin, Phys. Rev. 34, 999 (1929); E. A. Hylleraas, Z.f. Phys. 71,
739 (1931); E. Majorana, Atti Accad. Lincei 13, 58 (1931); J. K. L. Mac
Donald, Proc. Roy. Soc. A136, 528 (1932). The method of James and
Coolidge has been applied to several excited states of the hydrogen molecule
by R. D. Present, J. Chem. Phys. 3, 122 (1935), and by H. M. James, A. S.
Coolidge, and R. D. Present, in a paper to be published soon.
354 THE STRUCTURE OF SIMPLE MOLECULES [XH486
normal state of the molecule (Sec. 436, Weinbaum), and the other
an excited state. The term symbol ^^ for these states contains
the letter 2 to show that there is no component of electronic
orbital angular momentum along the nuclear axis; the superscript
I to show that the molecule is in a singlet state, as shown also
by the symbol S B , meaning symmetric in the positional coordi
nates of the two electrons, Pauli's principle then requiring that
the electronspin function b6 the singlet function
a(l)0(2) /9(l)a(2);
and the superscript + to show that the electronic wave function
is symmetric in the two nuclei, as shown also by S N . In
addition the subscript g (German gerade) is given to show that
the electronic wave function is an even function of the electronic
coordinates. Functions III and IV are both antisymmetric in
the nuclei, as indicated by the symbol A N and the superscript — ,
and are odd functions, as shown by the subscript u (German
ungerade), III being a triplet and IV a singlet function. A
further discussion of these symmetry properties will be given in
the next section and in Section 48.
Function III represents the repulsive interaction of two
normal hydrogen atoms, as mentioned in Section 43a. Function
II is mainly ionic in character and function IV completely so,
representing the interaction of H+ and H~\ Of these IV cor
responds to a known state, the first electronically excited state
of the molecule. As might have been anticipated from the
ionic character of the wave function, the state differs in its prop
erties from the other known excited states, having r, = 1.29 A
and v = 1358 cm 1 , whereas the others have values of r e and v e
close to those for the normal hydrogen moleculeion, 1.06 A
and 2250 cm  " 1 . The calculations of Zener and Guillemin and
of Hylleraas have shown that at the equilibrium distance the
wave function for this state involves some contribution from
wave functions for one normal and one excited atom (with
n = 2, 1 = 1), and with increase in r A B this contribution increases,
the molecule in this state dissociating into a normal and an
excited atom.
The state corresponding to II has not yet been identified.
Problem 481. Construct a wave function of symmetry type A N S*
from 1* and 2p functions.
Xn43f] THE HYDROGEN MOLECULE 355
43f. Oscillation and Rotation of the Molecule. Ortho and
Para Hydrogen. — In accordance with the discussion of the pre
ceding sections and of Chapter X, we can represent the complete
wave function for the hydrogen molecule as the product of
five functions, one describing the orbital motion of the electrons,
the second the orientation of electron spins, the third the oscilla
tional motion of the nuclei, the fourth the rotational motion of
the nuclei, and the fifth the orientation of nuclear spins (assuming
them to exist) :
(electronic\ / electronic \ , , w . ./ nuclear \
, . , y Y nuclear \/ nuclear \# \
A . \ .. jl oscillation I l rotation )\ . . .. }
motion /\onentation/\ /\ / ^orientation /
For the normal electronic state the first of these is symmetric
in the two electrons, the second antisymmetric, and the remaining
three independent of the electrons (and hence symmetric),
making the entire function antisymmetric in the two electrons,
as required by Pauli's principle. Let us now consider the sym
metry character of these functions with respect to the nuclei.
The first we have seen to be symmetric in the nuclei. The
second is also symmetric, not being dependent on the nuclear
coordinates. The third is also symmetric for all oscillational
states, inasmuch as the variable r which occurs in the oscilla
tional wave function is unchanged by interchanging the nuclei.
The rotational function, however, may be either symmetric or
antisymmetric. Interchanging the two nuclei converts the
polar angle # into t — # and <p into w + <P) the consideration of
the rotational wave functions (Sees. 35a and 21) shows that
this causes a change in sign if the rotational quantum number K
is odd, and leaves the function unchanged if K is even. Hence
the rotational wave function is symmetric in the nuclei for even
rotational states and antisymmetric for odd rotational states.
The nuclearspin function can be either symmetric or antisym
metric, providing that the nuclei possess spins.
•By an argument identical with that given in Section 296 for
the electrons in the helium atom we know that a system con
taining two identical protons can be represented either by wave
functions which are symmetric in the protons or by wave func
tions which are antisymmetric in the protons. Let us assume
356 THE STRUCTURE OF SIMPLE MOLECULES [XII43f
that the protons possessed no spins and that the symmetric
functions existed in nature. Then only the even rotational
states of the normal hydrogen molecule would occur (and only
the odd rotational states of the A N electronic state IV of the
preceding section). Similarly, if the antisymmetric functions
existed in nature, only the odd rotational states of the normal
molecule would occur. If, on the other hand, the protons
possessed spins of % (this being the value of the nuclearspin
quantum number /), both even and odd rotational states would
occur, in the ratio of 3 to 1 if the complete wave function were
symmetric or 1 to 3 if it were antisymmetric, inasmuch as there
are for I = }i three symmetric nuclearspin wave functions,
a(A) a(B),
±={a(A)P(B)+P(A)a{B)},
and
KA) KB),
and one antisymmetric one,
^={«(A) 8(B) j8(A)a(fl)}.
In this case, then, we would observe alternating intensities in
the rotational fine structure of the hydrogen bands, with the
ratio of intensities 3:1 or 1:3, depending on the symmetry char
acter of protons. Similar alternating intensities result from
larger values of /, the ratio being 1 / + 1 to /. It is seen that
1 Thus for / = 1 there are three spin functions for one particle, a, /3, and 7,
say, corresponding to m/ = f 1, 0, — 1. From these we can build the
following wave functions for two particles, giving the ratio 2:1.
Symmetric Antisymmetric
*(A) a(B)
KA) fi(B)
7(A) 7(B)
'a(A) ^(B) + fi(A) a(B)} ~^=\a(A) /3(B)  /3(A) a(B)}
s/2 \/2
■4=a(A) 7(B) + 7(A) a(B)\ ~
V2 V2
^{0(A)7(B) + 7(A) /3(B)} U
V2 V2
\a(A) 7(B) + 7(A) a(B)\ W«(A) 7(B)  7(A) «(B) )
{/3(A) 7(B) 4 7(A) /3(B)} ^=0(A) 7(B)  7(A) 0(B)]
XII43f] THE HYDROGEN MOLECULE 357
from the observation and analysis of band spectra of molecules
containing two identical nuclei the symmetry character and the
spin of the nuclei can be determined.
It was found by Dennison 1 (by a different method — the study
of the heat capacity of the gas, discussed in Section 49e) that
protons (like electrons) have a spin of onehalf, and that the
allowed wave functions are completely antisymmetric in the
proton coordinates (positional plus spin). This last statement
is the exact analogue of the Pauli exclusion principle. 2
Each of the even rotational wave functions for the normal
hydrogen molecule is required by this exclusion principle to be
combined with the antisymmetric spin function, whereas each
of the odd rotational wave functions can be associated with the
three symmetric spin functions, giving three complete wave
functions. Hence on the average there are three times as many
complete wave functions for odd rotational states as for even, and
at high temperatures three times as many molecules will be in
odd as in even rotational states (Sec. 49e). Moreover, a molecule
in an odd rotational state will undergo a transition to an even
rotational state (of the normal molecule) only extremely rarely,
for such a transition would result only from a perturbation involv
ing the nuclear spins, and these are extremely small in magnitude.
Hence (as was assumed by Dennison) under ordinary circum
stances we can consider hydrogen as consisting of two distinct
molecular species, one, called para hydrogen, having the nuclear
spins opposed and existing only in even rotational states (for
the normal electronic state), and the other, called ortho hydrogen,
having the nuclear spins parallel and existing only in the odd
rotational states. Ordinary hydrogen is onequarter para and
threequarters ortho hydrogen.
On cooling to liquidair temperatures the molecules of para
hydrogen in the main go over to the state with K = and
those of ortho hydrogen to the state with K — 1, despite the
fact that at thermodynamic equilibrium almost all molecules
would be in the state with K = 0, this metastable condition
being retained for months. It was discovered by Bonhoeffer
1 D. M. Dennison, Proc. Roy. Soc. A115, 483 (1927).
2 The spins and symmetry nature for other nuclei must at present be
determined experimentally; for example, it is known that the deuteron has
/ = 1 and symmetric wave functions.
358 THE STRUCTURE OF SIMPLE MOLECULES [Xn44a
and Harteck, 1 however, that a catalyst such as charcoal causes
thermodynamic equilibrium to be quickly reached, permitting
the preparation of nearly pure para hydrogen. It is believed
that under these conditions the orthopara conversion is due to
a magnetic interaction with the nuclear spins, 2 and not to dis
sociation into atoms and subsequent recombination, inasmuch as
the reaction H2 + D 2 ^ 2HD is not catalyzed under the same
conditions. The conversion is catalyzed by paramagnetic sub
stances 3 (oxygen, nitric oxide, paramagnetic ions in solution), and
a theoretical discussion of the phenomenon has been published. 4
At higher temperatures the conversion over solid catalysts seems
to take place through dissociation and recombination.
44. THE HELIUM MOLECULEION Hef AND THE INTERACTION
OF TWO NORMAL HELIUM ATOMS
In the preceding sections we have discussed systems of two
nuclei and one or two electrons. Systems of two nuclei and three
or four electrons, represented by the helium moleculeion He£
and by two interacting helium atoms, respectively, are treated
in the following paragraphs. A discussion of the results obtained
for systems of these four types and of their general significance
in regard to the nature of the chemical bond and to the structure
of molecules will then be presented in Section 45.
44a. The Helium Molecule ion He£. — In treating the system
of two helium nuclei and three electrons by the variation method
let us first construct electronic wave functions by using only
hydrogenlike Is orbital wave functions for the two atoms, which
we may designate as uj. and u By omitting the subscripts Is for
the sake of simplicity. Four completely antisymmetric wave
functions can be built from these and the spin functions a and f$.
These are (before normalization)
ft
Ma(1)«(1) u A (l)p(l) u B (l)a(l)
u A (2)a(2) 11,(2)0(2) u B {2) «(2)
u A (Z) a(3) u A (Z) 0(3) ti B (3) a(3)
(441)
1 K. F. Bonhoeffbr and P. Harteck, Z.f. phys. Chem. B4, 113 (1929).
* K. F. Bonhoeffer, A. Farkas, and K. W. Rummbl, Z. f. phys. Chem.
B21, 225 (1933).
«L. Farkas and H. Sachssb, Z. /. phys. Chem. B23, 1, 19 (1933).
« E. Wigner, Z. /. phys. Chem. B23, 28 (1933).
XII44a] THE HELIUM MOLECULEION 359
and
^ii
u B (l)a(l) u M (l)fi(l) u A (l)«(l)
iij(2)a(2) Ub(2)j8(2) u A (2)a(2)
u B (%) a(3) 1*^(3) 0(3) iu(3) a(3)
(442)
and two other functions, ^m and ^iv, obtained by replacing
a by in the last column of these functions. It is seen that the
function ^i represents a pair of electrons with opposed spins on
nucleus A (as in the normal helium atom) and a single electron
with positive spin on nucleus B; this we might write as He:
•He + . Function ^n similarly represents the structure He + :He,
the nuclei having interchanged their roles. It is evident that
this system shows the same degeneracy as the hydrogen molecule
ion, and that the solution of the secular equation for \f/i and ^n
will lead to the functions yf/ s and ^a, the nuclearsymmetric and
nuclearantisymmetric combinations of ^i and ^n (their sum
and difference), as the best wave functions given by this approxi
mate treatment. The other wave functions ^ m and ^iv lead
to the same energy levels.
The results of the energy calculation 1 (which, because of its
similarity to those of the preceding sections, does not need to be
given in detail) are shown in Figure 441. It is seen that the
nuclearantisymmetric wave function $ A corresponds to repulsion
at all distances, whereas the nuclearsymmetric function ^ s
leads to attraction and the formation of a stable moleculeion.
That this attraction is due to resorance between the structures
He: He + and He + :He is shown by comparison with the energy
curve for fa or ^n alone, given by the dashed line in Figure 441.
We might express this fact by writing for the normal helium
moleculeion the structure He—He*, and saying that its stabil
ity is due to the presence of a threeelectron bond between the
two atoms.
The function yps composed of Is hydrogenlike orbital wave
functions with effective nuclear charge 2e leads to a minimum
in the energy curve at r e = 1.01 A and the value 2.9 v.e. for the
energy of dissociation D e into He + He 4 *. A more accurate
treatment 2 can be made by minimizing the energy for each value
l L. Paulino, /. Chem. Phys. 1, 56 (1933).
1 L. Pauling, loc. cit. The same calculation with Z' given the fixed value
1.8 was made by E. Majorana, Nuovo Cim. 8, 22 (1931).
360
THE STRUCTURE OF SIMPLE MOLECULES [XII44a
of Tab with respect to an effective nuclear charge Z'e. This
leads to r e = 1.085 A, D e = 2.47 v.e., and the vibrational
frequency v e = 1950 cm 1 , with Z' equal to 1.833 at the equilib
rium distance. A still more reliable treatment can be made
by introducing two effective nuclear charges Z'e and Z"e, one for
the helium atom and one for the ion, and minimizing the energy
with respect to Z f and Z". This has been done by Weinbaum, 1
10
5
I T
1 \
I \
\ \
\ \
\ \
\ \
\ \
\ \
\ \
\ N
V \
\a n
X. S
1^^^^
05
1.0
r*
AB
15
2.0A
Fig. 441. — Energy curves for attractive and repulsive states of Het The
dashed curve corresponds to a nonexistent state, resonance between He: He +
and He« + :He being excluded.
who obtained the values r e = 1.097 A, D e = 2.22 v.e., Z' = 1.734,
and Z" = 2.029. The results of these calculations are in good
agreement with the experimental values given by excited states
of the diatomic helium molecule (consisting of the normal mole
culeion and an outer electron), which are r e = 1.09 A, D e = 2.5
v.e., and v e = 1650 cm 1 .
It is of interest that the system of a helium nucleus and
a hydrogen nucleus and three electrons does not show the
degeneracy of functions fa and ^ n , and that in consequence the
interaction of a normal helium atom and a normal hydrogen
i S. Weinbaum, /. Chem. Phys. 3, 547 (1935).
XII44b] THE HELIUM MOLECULEION 361
atom corresponds to repulsion, as has been verified by approxi
mate calculations. 1
44b. The Interaction of Two Normal Helium Atoms.— We may
write for the wave function for the normal state of a system
consisting of two nuclei and four electrons the expression
+ N
u*(l)«(l) «a(1)j8(1) m*(1)«(1) «,(l)j8(l)
u A (2)a(2) «a(2)i8(2) u B (2)a{2) u B {2) 0(2)
u A (3)a(3) ujl(S)P(Z) u B (3)a(3) u*(3) 0(3)
u A (l)a(4) 1^(4)0(4) «*(4) a(4) w B (4) j8(4)
, (443)
in which u A and u B represent Is wave functions about nuclei
A and B, respectively, and iV is a normalizing factor. This wave
function satisfies Pauli's principle, being completely anti
symmetric in the four electrons. It is the only wave function
of this type which can be constructed with the use of the one
electron orbital functions u A and u B alone.
It was mentioned by Heitler and London in their first paper 1
that rough theoretical considerations show that two normal
helium atoms repel each other at all distances. The evaluation
of the energy for the wave function \p of Equation 443 with
iia and u B hydrogenlike Is wave functions with effective atomic
number Z' = 2 %6 was carried out by Gentile. 2 A more
accurate calculation based on a heliumatom wave function not
given by a single algebraic expression has been made by Slater, 3
who found that the interaction energy is given by the approxi
mate expression
2.43ft
W  W ° = 7.70 • 10 10 e ao ergs. (444)
This represents the repulsion which prevents the helium atoms
from approaching one another very closely. The weak attrac
tive forces which give rise to the constant a of the van der Waals
equation of state cannot be treated by a calculation of this type
based on unperturbed heliumatom wave functions. It will
be shown in Section 476 that the van der Waals attraction is
given approximately by the energy term — 1.41e 2 ^or — 0.607p
1 W. Heitler and F. London, Z. f. Phys. 44, 455 (1927).
2 G. Gentile, Z. f. Phys. 63, 795 (1930).
8 J. C. Slater, Phys. Rev. 32, 349 (1928).
362 THE STRUCTURE OF SIMPLE MOLECULES [XII45
ergs. The equilibrium interatomic distance corresponding to
this attraction term and the repulsion term of Equation 444
is 3.0 A, in rough agreement with the experimental value of
about 3.5 A for solid helium, showing that the theoretical calcu
lations are of the correct order of magnitude.
45. THE ONBBLBCTRON BOND, THE ELECTRONPAIR BOND,
AND THE THREEELECTRON BOND
In the preceding sections we have discussed systems containing
two nuclei, each with one stable orbital wave function (a Is
function), and one, two, three, or four electrons. We have found
that in each case an antisymmetric variation function of the
determinantal type constructed from atomic orbitals and
spin functions leads to repulsion rather than to attraction
and the formation of a stable molecule. For the fourelectron
system only one such wave function can be constructed, so that
two normal helium atoms, with completed K shells, interact with
one another in this way. For the other systems, on the other
hand, more than one function of this type can be set up (the two
corresponding to the structures H H+ and H + H for the hydro
gen moleculeion, for example); and it is found on solution
of the secular equation that the correct approximate wave
functions are the sum and difference of these, and that in each
case one of the corresponding energy curves leads to attraction
of the atoms and the formation of a stable bond. We call
the bonds involving two orbitals (one for each nucleus) and one,
two, and three electrons the oneelectron bond, the electronpair
bond, and the threeelectron bond, respectively.
The calculations for the hydrogen moleculeion, the hydrogen
molecule, and the helium moleculeion show that for these
systems the electronpair bond is about twice as strong a bond
(using the dissociation energy as a measure of the strength of a
bond) as the oneelectron bond or the threeelectron bond. 1 This
fact alone provides us with some explanation of the great impor
tance of the electronpair bond in molecular structure in general
and the subsidiary roles played by the oneelectron bond and the
threeelectron bond. 2
1 See, however, the treatment of Li"^ by H. M. James, /. Chem. Phys. 3,
9 (1936).
*L. Pauling, /. Am. Chem. Soc. 53, 3225 (1931).
XII46] THE ONEELECTRON BOND 363
There is a still more cogent reason for the importance of the
electronpair bond. This is the nature of the dependence of the
energy of the bond on the similarity or dissimilarity of the two
nuclei (or the two orbitals) involved. Using only two orbitals,
u A and u B , we can construct for the oneelectron system only the
two wave functions
fa = u A (l) «(1)
and
fa = u B (l) «(1)
(together with two others involving 0(1) which do not combine
with these and which lead to the same energy curves). These
correspond to the electronic structures A B+ and A + \B. If
A and B are identical (or if fa and fai correspond to the same
energy because of an accidental relation between the orbitals
and the nuclear charges) there is degeneracy, and the interaction
of fa and fa causes the formation of a stable oneelectron bond.
If this equality of the energy does not obtain, the bond is weak
ened, the bond energy falling to zero as the energy difference for
\pi and 1^11 becomes very large.
The threeelectron bond behaves similarly. The wave func
tions (Eqs. 441 and 442) are closely related to those for the
oneelectron system, and the bond energy similarly decreases
rapidly in magnitude as the energy difference for the two wave
functions increases. Hence, in general, we expect strong one
electron bonds and threeelectron bonds not to be formed
between unlike atoms.
The behavior of the electronpair bond is entirely different.
The principal degeneracy leading to bond formation is that
between the wave functions
and
fa =
Ml)«(l) Ub(1)j8(1)
14,(2) a(2) v«(2)j8(2)
tfc(l)j8(l) t*B(l)a(l)
u A (2)fi(2) ti B (2)a(2)
These correspond to the same energy value even when A and B
are not identical; hence there is just the same resonance stabiliz
ing an electronpair bond between unlike atoms as between like
atoms. Moreover, the influence of the ionic terms is such as to
364 THE STRUCTURE OF SIMPLE MOLECULES [XII45
introduce still greater stability as the nuclei become more unlike.
One of the ionic wave functions
*m = u A (l) u a (2)
and
tiv = u B (l) u b (2)
«(1) 18(1)
a(2) 0(2)
'«(1)18(1)
«(2) /8(2)
corresponding to the ionic structures A : ~ B+ and A + : B~~, becomes
more and more important (contributing more and more to the
normal state of the molecule) as one of the atoms becomes more
electronegative than the other, in consequence of the lowering
of the energy for that ionic function. This phenomenon causes
electronpair bonds between unlike atoms to be, in general,
somewhat stronger than those between like atoms. The dis
cussion of this subject has been in the main empirical. 1
It has been found possible to apply quantummechanical
methods such as those described in this chapter in the detailed
discussion of the electronic structure of polyatomic molecules
and of valence and chemical bond formation in general. Only
in a very few cases has the numerical treatment of polyatomic
molecules been carried through with much accuracy; the most
satisfactory calculation of this type which has been made is
that of Coolidge 2 for the water molecule. General arguments
have been presented 3 which provide a sound formal justification
for the postulates previously made by the chemist regarding the
nature of valence. It can be shown, for example, that one bond
of the types discussed in this section can be formed by an atom
for each stable orbital of the atom. Thus the firstrow elements
of the periodic system can form as many as four bonds, by using
the four orbitals of the L shell, but not more. This result and
other results 4 regarding the relative orientation of the bond
axes provide the quantummechanical basis for the conception
of the tetrahedral carbon atom. Special methods for the
l L. Pauling, /. Am. Chem. Soc. 54, 3570 (1932).
2 A. S. Coolidge, Phys. Rev. 42, 189 (1932).
* W. Heitler, Z. f. Phys. 47, 835 (1928), etc.; F. London, Z. f. Phys.
60, 24 (1928), etc.; M. Born, Z.f. Phys. 64, 729 (1930); J. C. Slater, Phys.
Rev. 38, 1109 (1931).
* J. C. Slater. Phys. Rev. 34, 1293 (1929); L. Pauling, /. Am. Chem. Soc,
53, 1367 (1931); J. H. Van Vleck, /. Chem. Phys. 1, 177 (1933), etc.
XII45] THE ONEELECTRON BOND 365
approximate treatment of the stability of very complex molecules
such as the aromatic hydrocarbons 1 have also been developed
and found to be useful in the discussion of the properties of these
substances. The already very extensive application of wave
mechanics to these problems cannot be adequately discussed
in the small space which could be allowed it in this volume.
1 E. Huckel, Z. f. Phys. 70, 204 (1931), etc.; G. Rumer, Gottinger Nachr.
p. 337, 1932; L. Pauling, J. Chem. Phys. 1, 280 (1933); L. Pauling and
G. W. Wheland, ibid. 1, 362 (1933); L. Pauling and J. Sherman, ibid. 1,
679 (1933), etc.
CHAPTER XIII
THE STRUCTURE OF COMPLEX MOLECULES
In carrying out the simple treatments of the hydrogen mole
culeion, the hydrogen molecule, the helium moleculeion,
and the system composed of two normal helium atoms discussed
in the last chapter, we encountered no difficulty in constructing a
small number of properly antisymmetric approximate wave
functions out of oneelectron orbital functions for the atoms of
the molecule. The same procedure can be followed for more
complex molecules; it is found, however, that it becomes so
complicated as to be impracticable for any but the simplest
molecules, unless some method of simplifying and systematizing
the treatment is used. A treatment of this type, devised by
Slater, 1 is described in the following sections, in conjunction
with the discussion of a special application (to the system of
three hydrogen atoms). Slater's treatment of complex mole
cules has been the basis of most of the theoretical work which has
been carried on in this field in the last three years.
46. SLATER'S TREATMENT Ot COMPLEX MOLECULES
In the last chapter we have seen that a good approximation
to the wave function for a system of atoms at a considerable
distance from one another is obtained by using singleelectron
orbital functions u a (l), etc., belonging to the individual atoms,
and combining them with the electronspin functions a and p
in the form of a determinant such as that of Equation 443.
Such a function is antisymmetric in the electrons, as required by
Pauli's principle, and would be an exact solution of the wave
equation for the system if the interactions between the electrons
and those between the electrons of one atom and the nuclei
of the other atoms could be neglected. Such determinantal
1 J. C. Siatbb. Phya. Rev. 38, 1109 (1931).
366
XIII46] SLATER'S TREATMENT OF COMPLEX MOLECULES 367
functions are exactly analogous to the functions 1 used in Section
30a in the treatment of the electronic structure of atoms.
It may be possible to construct for a complex molecule many
such functions with nearly the same energy, all of which would
have to be considered in any satisfactory approximate treatment.
Thus if we consider one atom to have the configuration ls 2 2« 2 2p,
we must consider the determinantal functions involving all three
2p functions for that atom. A system of this type, in which
there are a large number of available orbitals, is said to involve
orbital degeneracy. Even in the absence of orbital degeneracy,
the number of determinantal functions to be considered may be
large because of the variety of ways in which the spin functions
a and can be associated with the orbital functions. This
spin degeneracy has been encountered in the last chapter; in the
simple treatment of the hydrogen molecule we considered the
two functions corresponding to associating positive spin with
the orbital u A and negative spin with u B , and then negative spin
with u A and positive spin with u B (Sec. 45). The four wave
functions described in Section 44a for the helium moleculeion
might be represented by the scheme of Table 461. The plus
Table 461. — Wave Functions for the Helium Moleculbion, .HbJ"
Function
u A
UB
2m.
I
+ 
+
+K
II
+
+ 
+H
III
+ 
—
H
IV
—
+ 
X
and minus signs show which spin function a or is to be asso
ciated with the orbital functions u A and u B (in this case Is func
tions on the atoms A and B, respectively) in building up the
determinantal wave functions. Thus row 1 of Table 461
corresponds to the function fa given in Equation 441.
The column labeled Sm a has the same meaning as in the atomic
problem; namely, it is the sum of the zcomponents of the spin
angular momentum of the electrons (with the factor h/2nr) m
Just as in the atomic case, the wave functions which have different
1 In Section 30a the convention was adopted that the symbol Ua(i) should
include the spin function a(i) or 0(i). In this section we shall not use the
convention, instead writing the spin function a or explicitly each time.
368
THE STRUCTURE OF COMPLEX MOLECULES [XIII46a
values of 2ra, do not combine with one another, so that we were
justified in Section 44a in considering only ^ r and \p n .
Problem 4G1. Set up tables similar to Table 461 for the hydrogen
molecule using the following choices of orbital functions: (a) Is orbitals
on the two atoms, allowing only one electron in each, (b) The same
orbitals but allowing two electrons to occur in a single orbital also; i.e.,
allowing ionic functions, (c) The same as (a) with the addition of func
tions 2p, on each atom, (d) The molecular orbital (call it u) obtained by
the accurate treatment of the normal state of the hydrogen moleculeion.
46a. Approximate Wave Functions for the System of Three
Hydrogen Atoms. — In the case of three hydrogen atoms we can
set up a similar table, restricting ourselves to the three Is func
tions w a , u b} and u c on three atoms a, b, and c, respectively, and
neglecting ionic structures (Table 462).
Table 462. — Wave Functions for the System of Three Hydrogen
Atoms
Function
U a
Ub
Uc
Sm,
I
+
+
+
+ «
II
+
+

43^
III
+

+
+H
IV

+
+
■VYi
V
+
~

 l A
VI

+

~K
VII
—

+
H
VIII
—
—
—
H
The wave function corresponding to row II of Table 462 is, for
illustration,
^ii =
1
VsH
U a (l)a(l) u b (l)a(l) u e (l)0(l)
u (2) a(2) u h (2) a(2) u c (2) fi(2)
u«(3) a(3) u b (3) a(3) u e (3) 0(3)
(461)
Each of the functions described in Table 462 is an approxi
mate solution of the wave equation for three hydrogen atoms;
it is therefore reasonable to consider the sum of them with
undetermined coefficients as a linear variation function. The
determination of the coefficients and the energy values then
requires the solution of a secular equation (Sec. 26d) of eight
rows and columns, a typical element of which is
#m  AinW (462)
XHI46b] SLATER'S TREATMENT OF COMPLEX MOLECULES 369
where
#m « JWiMr, (463)
and
Ai ir = jWudr, (464)
H being the complete Hamiltonian operator for the system.
Problem 462. Make a table similar to Table 462 but including all
ionic functions that can be made with the use of u a , Ub, and u e .
46b. Factoring the Secular Equation. — In the discussion of the
electronic structure of atoms (Sec. 30c) we found that the
secular equation could be factored to a considerable extent
because integrals involving wave functions having different
values of Sm, or different values of 2m z (the .quantum numbers
of the components of spin and orbital angular momentum,
respectively) are zero. In the molecular case the orbital angular
momentum component is no longer a constant of the motion
(Sec. 52), so that only the spin quantum numbers are useful
in factoring the secular equation.
In the case of the system under discussion, we see from Table
462 that the secular equation factors into two linear factors
(2ra, = % and —%) and two cubic factors (2w, = Yi
and — J^). On the basis of exactly the Same reasoning as used
in Section 30c for the atomic case, we conclude that the roots of
the two linear factors will be equal to each other and also to
one of the roots of each of the cubic factors. 1 The four cor
responding wave functions are therefore associated with a quartet
energy level, which on the vector picture corresponds to the
parallel orientation of the three spin vectors, the four states
differing only in the orientation of the resultant vector.
The two remaining energy levels will occur twice, once in
each of the cubic factors. Each of them is, therefore, a doublet
level. The straightforward way of obtaining their energy values
would be to solve the cubic equation; but this is unnecessary,
inasmuch as by taking the right linear combinations of II, III,
and IV it is possible to factor the cubic equation into a linear
factor and a quadratic factor, the linear factor yielding the
energy of the quartet level. Such combinations are
1 These statements can easily be verified by direct comparison of the
roots obtained, using the expressions for the integrals given in Section 46c.
370
THE STRUCTURE Of COMPLEX MOLECULES [XHI46c
.1 =
vf
(II  III),
B
= t=(III
V2
IV),
and
c  ^(iv  id,
Z> = L(II + III + IV).
(465)
(466)
(467)
(468)
Since these four functions are constructed from only three linearly
independent functions II, III, and IV, they cannot be linearly
independent; in fact, it is seen that A + B + C = 0. The
factoring of the secular equation will be found to occur when it
is set up in terms of D and any two of the functions A, B f and C.
The energy of the quartet level can be obtained from either of
the linear factors; it is given by the relation
An
(469)
The values of the energy of the two doublet levels are obtained
from the quadratic equation
H AA  AaaW Hab  AabW
Hba ~ &baW Ebb  &bbW
o,
(4610)
in which
Hab = JA*HBdT,)
Aab  JA*Bdr, >
(4611)
Problem 463. Indicate how the secular equation for each of the cases
of Problem 461 will factor by drawing a square with rows and columns
labeled by the wave functions which enter the secular equation, and indi
cating by zeros in the proper places in the square the vanishing matrix
elements.
46c. Reduction of Integrals. — Before discussing the conclusions
which can be drawn from these equations, let us reduce somewhat
further the integrals H uulf etc. The wave function II can be
written in the form (Sec. 30a)
fct = ^2("l) p Nl)a(lW2)a(2) Mc (3)^(3),
p
(4612)
Xm46c] SLATER'S TREATMENT OF COMPLEX MOLECULES 371
in which P represents a permutation of the functions u a a t etc.,
among the electrons. A typical integral can thus be expressed
In the form
Hum  2 2 ( ~ !)^^ f p ^ * (1)a(1)u * (2)a(2)w *< 3 >^( 3 > H
P , P
Pu a (l)a(l)u b (2)P(2)u e (3)a(3)dr. (4613)
Following exactly the argument of Section 30d for the atomic
case, we can reduce this to the form
Hum  2(l)W(l)"(l)<(2)a(2)^^
p
ii»(2)0(2)u«(3)a(3)dr. (4614)
As in the atomic case, the integral vanishes unless the spins
match, and there can be no permutation P which matches the
spins unless 2m, is the same for II and III. In this case we
see that the spins are matched for the permutations P which
permute 123 into 132 or 231 so that only these terms contribute
to the sum. When the spins match in an integral, the integration
over the spin can be carried out at once, yielding the factor
unity. We thus have the result
K*(l) «(1) <(2) a(2) u c *(3) 0(3) J5M1) «(1) ti»(3) 0(3) u c {2) «(2)
dr = /u*(l) w 6 *(2) u*(3) Hu a {\) u c (2) ti»(3) dr = (abc\H \acb),
(4615)
in which we have introduced a convenient abbreviation,
(abc\H\acb).
In this way we obtain the following expressions:
Hu  (abc\H\abc)  (abc\H\bac)  (abc\H\acb)
 (abc\H\cba) + (abc\H\bca) + (dbc\H\cab), ]
ffim  (abc\H\abc)  (abc\H\bac),
Hmm  (a6cHo6c)  (akff<*a),
Hiviv  (abcffa6c)  (abcffac&), / < 46 ~ 16 '
Hum  (abc\H\cab)  (a6ci?ac6),
Hmiv  (abc\H\cab)  (a6ctf6ac),
Hinv  (abc\H\bca)  (a6cffc&a).
The expressions for the A's are the same with H replaced by
unity. The integral (dbc\H\abc) is frequently called the Coulomb
integral, because it involve? the Coulomb interaction of two
372 THE STRUCTURE OF COMPLEX MOLECULES [XIII46d
distributions of electricity determined by u a , Ub t and u c . The
other integrals such as (abc\H\bac) are called exchange integrals.
If only one pair of orbitals has been permuted, the integral is
called a single exchange integral; if more than one, a multiple
exchange integral. If the orbital functions u a , u h} and u c were
mutually orthogonal, many of these integrals would vanish, but
it is seldom convenient to utilize orthogonal orbital functions in
molecular calculations. Nevertheless, the deviation from orthog
onality may not be great, in which case many of the integrals
can be neglected.
46d. Limiting Cases for the System of Three Hydrogen Atoms.
The values of the integrals #imi, etc., depend on the distances
between the atoms a, 6, and c, and therefore the energy values
and wave functions will also depend on these distances. It is
interesting to consider the limiting case in which a is a large dis
tance from b and c, which are close together. It is clear that the
wave function u a will not overlap appreciably with either u h or
u ey so that the products u a Ub and u a u c will be essentially zero for
all values of the coordinates. Such integrals as (abc\H\bac)
will therefore be practically zero, and we can write
#iui = #111111 = (abc\H\abc),
Hui iv = #n iv = 0,
#iviv = (dbc\H\abc) — (abc\H\acb),
and
#iim = ~ (abc\H\acb),
thus obtaining the further relations
H AA = (abc\H\abc) + (abc\H\acb),
H BB = (abc\H\abc)  Y 2 {abc\H\acb) ,
and
Has = y 2 (abc\H\abc)  y 2 {abc\H\acb) .
If we insert these values into the secular equation 4610 we obtain
as one of the roots the energy value
W = ^> (4617)
and we find that the corresponding wave function is just the
function A itself.
It is found on calculation that exchange integrals involving
orbitals on different atoms are usually negative in sign. In
XIII46d] SLATER'S TREATMENT OF COMPLEX MOLECULES 373
case that such an integral occurs in the energy expression with a
positive coefficient, it will contribute to stabilizing the molecule
by attracting the atoms toward one another. Thus the expres
sion for H A a includes the Coulomb integral (abc\H\abc) and the
exchange integral (abc\H\acb) with positive coefficient. Hence
atoms b and c will attract one another, in the way corresponding
to the formation of an electronpair bond between them (exactly
as in the hydrogen molecule alone). Similarly the function B
represents the structure in which atoms a and c are bonded, and
C that in which a and b are bonded.
When we bring the three atoms closer together, so that all the
interactions are important, none of these functions alone is the
correct combination; they must be combined to give a wave
function which represents the state of the system. Therefore
when three hydrogen atoms are near together, it is not strictly
correct to say that a certain two of them are bonded, while the
third is not.
We can, however, make some statements regarding the
interaction of a hydrogen molecule and a hydrogen atom on the
basis of the foregoing considerations. We have seen that when
atom a is far removed from atoms b and c (which form a normal
hydrogen molecule), the wave function for the system is function
A. As a approaches b and c the wave function will not differ
much from A, so long as the ab and ac distances are considerably
larger than the be distance. An approximate value for the
interaction energy will thus be H A a/&aa, with
Haa — liiHnu + Hniui — 2Huiu)
= (abc\H\abc) + (abc\H\acb)
y 2 (abc\H\bac)  V 2 {abc\H\cba)  (abc\H\cab),
and a similar expression for Aaa It is found by calculation that
in general the single exchange integrals become important at
distances at which the Coulomb integral and the orthogonality
integral have not begun to change appreciably, and at which
the multiple exchange integrals [(abc\H\cab) in this case] are still
negligible. Thus we see that the interaction energy of a hydro
gen atom and a hydrogen molecule at large distances is
y 2 (abc\H\bac)  y 2 (abc\H\cba) .
Each of these terms corresponds to repulsion, showing that the
molecule will repel the atom.
374 THE STRUCTURE OF COMPLEX MOLECULES [XIII46e
Approximate discussions of the interaction of a hydrogen atom
and hydrogen molecule have been given by Eyring and Polanyi, 1
and a more accurate treatment for some configurations has been
carried out by Coolidge and James. 2
46e. Generalization of the Method of Valencebond Wave
Functions. — The procedure which we have described above for
discussing the interaction of three hydrogen atoms can be
generalized to provide an analogous treatment of a system con
sisting of many atoms. Many investigators have contributed to
the attack on the problem of the electronic structure of complex
molecules, and several methods of approximate treatment have
been devised. In this section we shall outline a method of treat
ment (due in large part to Slater) which may be called the
method of valencebond wave functions, without giving proofs of
the pertinent theorems. The method is essentially the same as
that used above for the threehydrogenatom problem.
Let us now restrict our discussion to the singlet states of
molecules with spin degeneracy only. For a system involving
2n electrons and 2n stable orbitals (such as the Is orbitals in 2n
hydrogen atoms), there are (2n)!/2 n n! different ways in which
valence bonds can be drawn between the orbitals in pairs. Thus
for the case of four orbitals a, 6, c, and d the bonds can be drawn
in three ways, namely,
a b a b a b
d c d c d c
A B C
There are, however, only t , /'m independent singlet wave
functions which can be constructed from the 2n orbitals with
one electron assigned to each Orbital (that is, with neglect of
ionic structures). It was shown by Slater that wave functions
can be set up representing structures A, B, and C, and that only
two of them are independent. The situation is very closely
analogous to that described in Section 466.
1 H. Eyring and M. Polanyi, Naturwissenschaften, 18, 914 (1930);
Z.f. phys. Chem. B12, 279 (1931).
2 A. S. Coolidge and H. M. Jambs, /. Chem. Phys., 2, 811 (1934).
Xin46e] SLATER'S TREATMENT OF COMPLEX MOLECULES 375
The very important observation was made by Rumer 1 that
if the orbitals a, b } etc. are arranged in a ring or other closed
concave curve (which need have no relation to the nuclear con
figuration of the molecule), and lines are drawn between orbitals
bonded together (the lines remaining within the closed curve),
the structures represented by diagrams in which no lines intersect
are independent. These structures are said to form a canonical
set. Thus in the above example the canonical set (correspond
ing to the order a, 6, c, d) comprises structures A and B. For
six orbitals there are five independent structures, as shown
in Figure 461.
N* <S
n
m
m
%,
<^
mis:
Fig. 461. — The five canonical valencebond structures for six orbitals, and
some of their superposition patterns.
The wave function corresponding to the structure in which
orbitals a and 6, c and d, etc. are bonded is
Po(l) 0(1) 6(2) «(2) c(3) 0(3) d(4) a(4)
]•
(4618?
in which P is the permutation operator described above (Sec.
46c), and R is the operation of interchanging the spin functions
a and p of bonded orbitals, such as a and b. The factor ( — 1) R
equals +1 for an even number of interchanges and —1 for an
odd number. The convention is adopted of initially assigning
the spin function to orbital a, a to 6, etc.
1 G. Rumer, Gottinger Nachr., p. 377, 1932.
376 THE STRUCTURE OF COMPLEX MOLECULES [XIII46e
A simple method has been developed 1 of calculating the coeffi
cients of the Coulomb and exchange integrals in setting up the
secular equation. To find the coefficient of the Coulomb
integral for two structures, superimpose the two bond diagrams,
as shown in Figure 461. The superposition pattern consists
of closed polygons or islands, each formed by an even number of
bonds. The coefficient of the Coulomb integral is 3^ n ~\ where
i is the number of islands in the superposition diagram. Thus
we obtain Hu = Q + • • • , Hm = }4Q + • • • , etc., in which
Q represents the Coulomb integral (abed • • • \H\abcd • • • ).
The coefficient of a single exchange integral such as
(ab) = (abed • • • \H\bacd • • • )
is equal to //2 n ~*, in which /has the value — 3^ if the two orbitals
involved (a and b) are in different islands of the superposition
pattern; +1 if they are in the same island and separated by an
odd number of bonds (along either direction around the polygon) ;
and —2 if they are in the same island and separated by an even
number of bonds. Thus we see that
ffn = Q  }i(ac) + (ab) + • • • , H in = Q  2(ac) + (ab) +
• • • , etc.
Let us now discuss the energy integral for a particular valence
bond wave function, in order to justify our correlation of valence
bond distribution and wave function as given in Equation 4618.
The superposition pattern for a structure with itself, as shown
by I I in Figure 461, consists of n islands, each consisting of
two bonded orbitals. We see that
pn _ ffii __ J_//i I "S^ /single exchange integrals for bonded\
An Au\ ^J\ pairs of orbitals /
__ j/ %^ /single exchange integrals for nonbonded\
/2 ^— J \ pairs of orbitals /
+ higher exchange integrals?. (4619)
It is found by calculation that the single exchange integrals
are as a rule somewhat larger in magnitude than the other
integrals. Moreover, the single exchange integral for two orbitals
l L. Pauling, J. Chem. Phys. 1, 280 (1933). See also H. Eyring and
G. E. Kimball, J. Chem. Phys. 1, 239 (1933), for another procedure.
XIII46f] SLATER'S TREATMENT OF COMPLEX MOLECULES 377
on different atoms is usually negative in value for the interatomic
distances occurring in molecules, changing with interatomic
distance in the way given by a Morse curve (Sec. 35d). Those
single exchange integrals which occur with the coefficient +1 in
Equation 4619 consequently lead to attraction of the atoms
involved in the exchange, and the other single exchange integrals
(with coefficient — }4) l ea d to repulsion; in other words, the
wave function corresponds to attraction of bonded atoms and
repulsion of nonbonded atoms and is hence a satisfactory wave
function to represent the valencebond structure under discussion.
The valencebond method has been applied to many problems,
some of which are mentioned in the following section. It has
been found possible to discuss many of the properties of the
chemical bond by approximate wavemechanical methods; an
especially interesting application has been made in the treatment
of the mutual orientation of directed valence bonds, 1 leading to
the explanation of such properties as the tetrahedral orientation
and the equivalence of the four carbon valences.
46f. Resonance among Two or More Valencebond Struc
tures. — It is found that for many molecules a single wave function
of the type given in Equation 4618 is a good approximation to
the correct wave function for the normal state of the system j
that is, it corresponds closely to the lowest root of the secular
equation for the spindegeneracy problem. To each of these
molecules we attribute a single valencebond structure, or
electronic structure of the type introduced by G. N. Lewis, with
two electrons shared between two bonded atoms, as representing
satisfactorily the properties of the molecule.
In certain cases, however, it is evident from symmetry or other
considerations that more than one valencebond wave function is
important. For example, for six equivalent atoms arranged at
the corners of a regular hexagon the two structures I and II of
Figure 461 are equivalent and must contribute equally to the
wave function representing the normal state of the system.
It can be shown that, as an approximation, the benzene molecule
can be treated as a sixelectron system. Of the total of 30 valence
electrons of the carbon and hydrogen atoms, 24 can be considered
1 J. C. Slater, Phys. Rev. 37, 481 (1931); L. Pauling, J. Am. Chem. Soc. 63,
1367 (1931); J. H. Van Vleck, /. Chem. Phys. 1, 177 (1933); R. Hultgrbn,
Phys. Rev. 40, 891 (1932).
378
THE STRUCTURE OF COMPLEX MOLECULES [XIII 46f
to be involved in the formation of single bonds between adjacent
atoms, giving the structure
H
Hs
C
I
c
H
/
V
H
^H
H
These single bonds use the Is orbital for each of the hydrogen
atoms and three of the L orbitals for each carbon atom. There
remain six L orbitals for the carbon atoms and six electrons, which
can be represented by five independent wave functions corre
sponding to the five structures of Figure 461. We see that
structures I and II are the Kekule* structures, with three double
bonds between adjacent atoms, whereas the other structures
involve only two double bonds between adjacent atoms. If,
as an approximation, we consider only the Kekule* structures,
we obtain as the secular equation
Hi i  AnW #m  AiuW
Hiu  &inW Hnu — kniiW
in which also H u u = Hn and An n = Ai i.
The solutions of this are
Hi i + Hi n
0,
and
W =
W =
An + Ai ii
Hi i — Hi ii
An
Am
the corresponding wave functions being ^i f ^ n and ^i — ^ n .
Thus the normal state of the system is more stable than would
correspond to either structure I or structure II. In agreement
with the discussion of Section 41, this energy difference is called
the energy of resonance between the structures I and II.
As a simple example let us discuss the system of four equivalent
univalent atoms arranged at the corners of a square. The two
structures of a canonical set are
XIII 46f J SLATER'S TREATMENT OF COMPLEX MOLECULES 379
a b
a b
and
d c
I
d c
II
If we neglect all exchange integrals of H except the single
exchange integrals between adjacent atoms, which we call
a [a = (ab) = (be) = (cd) = {da)}, and all exchange integrals
occurring in A, the secular equation is found by the rules of
Section 46e to be
= 0.
Q + aW 1 AQ + 2ay 2 W
y 2 Q + 2a y 2 w Q + a  w
The solutions of this are W = Q + 2a and W = Q — 2a, of
which the former represents the normal state, a being negative
in sign. The energy for a single structure (I or II) is Wi = Q + a;
hence the resonance between the two structures stabilizes the
system by the amount a.
Extensive approximate calculations of resonance energies foi
molecules, especially the aromatic carbon compounds, have
been made, and explanations of several previously puzzling
phenomena have been developed. 1 Empirical evidence has
also been advanced to show the existence of resonance among
several valencebond structures in many simple and complex
molecules. 2
It must be emphasized, as was done in Section 41, that the use
of the term resonance implies that a certain type of approximate
treatment is being used. In this case the treatment is based
on the valencebond wave functions described above, a procedure
which is closely related to the systematization of molecule
formation developed by chemists over a long period of years,
and the introduction of the conception of resonance has per
mitted the valencebond picture to be extended to include
1 E. Huckel, Z. f. Phys. 70, 204 (1931), etc.; L. Pauling and G. W.
Wheland, /. Chem. Phys. 1, 362 (1933); L. Pauling and J. Sherman, ibid.
1, 679 (1933); J. Sherman, ibid. 2, 488 (1934); W. G. Penney, Proc. Roy.
Soc. A146, 223 (1934); G. W. Wheland, J. Chem. Phys. 3, 230 (1935).
2 L. Pauling, /. Am. Chem. Soc. 54, 3570 (1932); Proc. Nat. Acad. Sci.
18, 293 (1932); L. Pauling and J. Sherman, /. Chem. Phys. 1, 606 (1933);
G. W. Wheland, ibid, 1, 731 (1933); L. O. Brockway and L. Pauling,
Proc. Nat. Acad. Sci. 19, 860 (1933).
380 THE STRUCTURE OF COMPLEX MOLECULES [XIH46g
previously anomalous cases. A further discussion of this point
is given in the following section.
Problem 464. Set up the problem of resonance between three equivalent
structures or functions fa, fai, ^m, assuming that Hn = Him = #111 in,
etc. Solve for the energy levels and correct combinations, putting Ai i = 1
and Ai n = 0.
Problem 465. Evaluate the energy of a benzene molecule, considered
as a sixelectron problem: (a) considering only one Kekul6 structure;
(b) considering both Kekule* structures; (c) considering all five structures.
Neglect all exchange integrals of H except
(ab) = (be) = (cd) = (de) = (ef) = (fa) = a,
and all exchange integrals entering in A.
46g. The Meaning of Chemical Valence Formulas. — The
structural formulas of the organic chemist have been determined
over a long period of years as a shorthand notation which
describes the behavior of the compound in various reactions,
indicates the number of isomers, etc. It is only recently that
physical methods have shown directly that they are also fre
quently valid as rather accurate representations of the spatial
arrangement of the atoms. The electronic theory of valence
attempted to burden them with the additional significance of
maps of the positions of the valence electrons. With the advent
of quantum mechanics, we know that it is not possible to locate
the electrons at definite points in the molecule or even to specify
the paths on which they mov6. However, the positions of maxi
mum electron density can be calculated, and, as shown in Figure
424, the formation of a bond does tend to increase the electron
density in the region between the bonded atoms, which therefore
provides a revised interpretation of the old concept that the
valence electrons occupy positions between the atoms.
The discussion of Section 466 shows that, at least in certain
cases, the valencebond picture can be correlated with an approxi
mate solution of the wavemechanical problem. This correla
tion, however, is not exact in polyatomic molecules because
functions corresponding to other ways of drawing the valence
bonds also enter, although usually to a lesser extent.
Thus the valence picture may be said to have a definite signifi
cance in terms of wave mechanics in those cases in which one
valencebond wave function is considerably more important than
the others, but where this is not true the significance of the
Xm46h] SLATER'S TREATMENT OF COMPLEX MOLECULES 381
structural formulas is less definite. Such less definite cases are
those which can be described in terms of resonance. It is notable
that the deficiency of the single structural formula in such cases
has long been recognized by the organic chemist, who found that
no single formula was capable of describing the reactions and
isomers of such a substance as benzene. In a sense, the use of
the term resonance is an effort to extend the usefulness of the
valence picture, which otherwise is found to be an. imperfect
way of describing the state of many molecules.
46h. The Method of Molecular Orbitals. — Another method of
approximate treatment of the electronic structure of molecules,
called the method of molecular orbitals, has been developed and
extensively applied, especially by Hund, Mulliken, andHiickel. 1
This method, as usually carried out, consists in the approximate
determination of the wave functions (molecular orbitals) and the
associated energy values for one electron in a potential field corre
sponding to the molecule. The energy of the entire molecule is
then considered to be the sum of the energies of all the electrons,
distributed among the more stable molecular orbitals with no
more than two electrons per orbital (Pauli's principle). A
refinement of this method has been discussed in Section 436 in
connection with the hydrogen molecule.
As an example let us consider the system of four equivalent
univalent atoms at the corners of a square, discussed in the
previous section by the valencebond method. The secular
equation for a oneelectron wave function (molecular orbital),
expressed as a. linear combination of the four atomic orbitals
u a > u b} u C) and u d , is
= 0,
 W P p
P q  W P
p q  W p
P P qW
in which q is the Coulomb integral fu a (l) H'u a {l) dr and P is the
exchange integral /u a (l) H'u b (l) dr for adjacent atoms, H' being
the Hamiltonian operator corresponding to the molecular
1 F. Hund, Z.f. Phys. 73, 1, 565 (19311932); R. S. Mulliken, /. Chem.
Phys. 1, 492 (1933); etc.; J. E. LennardJones. Trans. Faraday Soc. 25,
668 (1929); E. Huckel, Z.f. Phys. 72, 310 (1931); 76, 628 (1932); 83, 632
(1933); Trans. Faraday Soc. 30, 40 (1934).
382 THE STRUCTURE OF COMPLEX MOLECULES [XIII46h
potential function assumed. We neglect all other integrals.
The roots of this equation are
Wi = q + 20,
W 2 = q,
Ws = q,
W 4 = q  20.
Since is negative, the two lowest roots are Wi and W 2 (or TT 3 ),
and the total energy for four electrons in the normal state is
W = 2Wi + 2W 2 = 4? + 40.
If there were no interaction between atoms a, 6 and c, d
(corresponding to bond formation allowed only between a and b
and between c and d), the energy for four electrons would still
be 4g + 40. Accordingly in this example the method of molecu
lar orbitals leads to zero resonance energy. This is in poor
agreement with the valencebond method, which gave the
resonance energy a. In most cases, however, it is found that
the results of the two methods are in reasonably good agreement,
provided that be given a value equal to about 0.6 a (for aromatic
compounds). A comparison of the two methods of treatment
has been made by Wheland. 1 It is found that the valencebond
method, when it can be applied, seems to be somewhat more
reliable than the molecularorbital method. On the other hand,
the latter method is the more simple one, and can be applied to
problems which are too difficult for treatment by the valence
bond method.
Problem 4S6. Treat the system of Problem 465 by the molecular
orbital method. Note that the resonance energy given by the two methods
is the same if /3 = 0.553 a (using part c of Problem 465).
1 G. W. Wheland, J. Chem. Phys. 2, 474 (1934).
CHAPTER XIV
MISCELLANEOUS APPLICATIONS OF QUANTUM
MECHANICS
In the following three sections we shall discuss four applications
of quantum mechanics to miscellaneous problems, selected from
the very large number of applications which have been made.
These are: the van der Waals attraction between molecules
(Sec. 47), the symmetry properties of molecular wave functions
(Sec. 48), statistical quantum mechanics, including the theory
of the dielectric constant of a diatomic dipole gas (Sec. 49),
and the energy of activation of chemical reactions (Sec. 50).
With reluctance we omit mention of many other important
applications, such as to the theories of the radioactive decomposi
tion of nuclei, the structure of metals, the diffraction of electrons
by gas molecules and crystals, electrode reactions in electrolysis,
and heterogeneous catalysis.
47. VAN DER WAALS FORCES
The first detailed treatments of the weak forces between
atoms and molecules known as van der Waals forces (which are
responsible for the constant a of the van der Waals equation of
state) were based upon the idea that these forces result from the
polarization of one molecule in the field of a permanent dipole
moment or quadrupole moment of another molecule, 1 or from
the interaction of the permanent dipole or quadrupole moments
themselves. 2 With the development of the quantum mechanics
it has been recognized (especially by London 3 ) that for most
molecules these interactions are small compared with another
interaction, namely, that corresponding to the polarization of
one molecule in the rapidly changing field due to the iiistan
1 P. Debye, Phys. Z. 21, 178 (1920); 22, 302 (1921).
2 W. H. Keesom, Proc. Acad. Sci. Amsterdam 18, 636 (1915); Phys. Z. 22,
129, 643 (1921).
« F. London, Z. f. Phys. 63, 245 (1930).
383
384 MISCELLANEOUS APPLICATIONS [XIV47a
taneous configuration of electrons and nuclei of another mole
cule; that is, in the main the polarization of one molecule by the
timevarying dipole moment of another. In the following sec
tions we shall discuss the approximate evaluation of the energy
of this interaction by variation and perturbation methods for
hydrogen atoms (Sec. 47a) and helium atoms (Sec. 476), and then
briefly mention the approximate semiempirical discussion for
molecules in general (Sec. 47c).
47a. Van der Waals Forces for Hydrogen Atoms. — For large
values of the internuclear distance r A B — R the exchange phenom
enon is unimportant, and we can take as the unperturbed wave
function for a system of two hydrogen atoms the simple product
of two hydrogenlike Is wave functions,
r = u U a(1) u U b(2). (471)
The perturbation for this function consists of the potential
energy terms
p2, p" pi, p"
H f = —  — + — + — • (472)
Tbi r A 2 r A B r l2
Now this expression can be expanded in a Taylor's series in
inverse powers of R = r A B, to give (with the two atoms located
on the z axis)
e 2 3 e 2
H' = ^g(xix 2 + 2/12/2  2ziz 2 ) + 2R*l r * z * "~ r * Zl
+ (2xiz 2 + 22/12/2 ~ 3siz 2 )(zi  z 2 )}
+ l^M r l ~ 6rUl  $r\z\  \bz\z\
+ 2(31*2 + 2/12/2 + 4z!Z 2 ) 2 J + • • • , (473)
in which x if 1/1, «i are coordinates of the first electron relative to
its nucleus, and x 2} 2/2, z 2 are coordinates of the second electron
relative to its nucleus. The first term represents the interaction
of the dipole moments of the two atoms, the second the dipole
quadrupole interaction, the third the quadrupolequadrupole
interaction, and so on.
Let us first consider only the dipoledipole interaction, using
the approximate secondorder perturbation treatment 1 of Section
1 The firstorder perturbation energy is zero, as can be seen from inspec
tion of the perturbation function.
XIV47a] VAN DER WAALS FORCES 385
27e. It is necessary for us to evaluate the integral
with H' given by
H' = j^faxi + yiy2  2z l z i ). (474)
It is seen that the crossproducts in (i/') 2 vanish on integration,
so that we obtain
£j>«
or
(ff'% = ~J ^*rJr,V«Wr = grfrj. (475)
This expression, with r\ and r^ replaced by their value 3a§ (Sec.
21c), gives, when introduced in Equation 2747 together with
Wq = —e 2 /a 0j the value for the interaction energy
W' Q ' = 5?JJ?. (476)
The fact that this value is also given by the variation method
with the variation function ^°(1 + AH') shows that this is an
upper limit for W'J (a lower limit for the coefficient of — e 2 al/R*).
Moreover, by an argument similar to that of the next to the
last paragraph of Section 27e it can be shown that the value
— 8p is a lower limit to W' ' f so that we have thus determined
the value of the dipoledipole interaction to within about 15
per cent.
Variation treatments of this problem have been given by Slater
and Kirkwood, 1 Hass6, 2 and Pauling and Beach. 3 It can be
easily shown 4 that the secondorder perturbation energy can
be obtained by the use of a variation function of the form
* = *°{l+ff , /(r 1 ,r t )},
with H' given by Equation 474. The results of the variation
1 J. C. Slater and J. G. Kirkwood, Phys. Rev. 37, 682 (1931).
* H. R. Hasse, Proc. Cambridge Phil Soc. 27, 66 (1931). A rough treatment
for various states has been given by J. Podolanski, Ann. d. Phys. 10, 695
(1931).
* L. Pauling and J. Y. Beach, Phys. Rev. 47, 686 (1935).
4 This was first shown by Slater and Kirkwood.
386
MISCELLANEOUS APPLICATIONS
[XIV47a
treatment for different functions /(ri, r 2 ) are given in Table 471.
It is seen that the coefficient of —e 2 a\/R* approaches a value 1
only slightly larger than 6.499; this can be accepted as very close
to the correct value.
So far we have considered only dipoledipole interactions.
Margenau 2 has applied the approximate secondorder perturba
tion method of Section 27c to the three terms of Equation 473,
obtaining the expression
W"
R*
R*
1416e 2 a 9 n
R l
+
• (477)
It is seen that the higherorder terms become important at small
distances.
Table 471.— Variation Treatment of van der Waals Interaction
of Two Hydrooen Atoms
Variation function u lsA (l) tti„»(2)<l + (xiz 2 + f/iJ/2  2z l z t )f(r i , r 2 )f
/(ri, r 2 )
E W°
Reference*
1. A
6.00e 2 a V^ 6
6 14
6 462
6 469
6 482
6 49
6 490
6 498
6 4984
6.49899
6.49903
H
2.  l Ar l r 2 /{r l \ r 2 )
SK
3. A +B(n + r 2 )
4. A + Brir 2
5. A + B(n + r 2 ) + C ri r 2 . .
6. Ar\r v 2 (v = 0.325)
7. A + Br x r 2 +Ci\r\
8. A + Br x r 2 + C/JrJ + 1) r\r\
9. Polynomial f to r\r\
10. Polynomial to rjr^
11. Polynomial to r\r\
PB
II
PB
SK
II
II
PB
PB
PB
* H = Hass6, SK = Slater and Kirkwood, PB  Pauling and Beach.
f The polynomial contains all terms of degree 2 or less in n and 2 or less in ri.
1 A straightforward but approximate application of secondorder perturba
tion theory by R. Eisenschitz and F. London gave the value 6.47 for this
coefficient [Z. /. Phys. 60, 491 (1930)]. The first attack on this problem
was made by S. C. Wang, Phys. Z. 28, 663 (1927). The value found by
him for the coefficient, 24 % 8 = 8.68, must be in error (as first pointed oul,
by Eisenschitz and London), being larger than the upper limit 8 given above.
The source of the error has been pointed out by Pauling and Beach, loc. cit.
2 H. Margenau, Phys. Rev. 38, 747 (1931). More accurate values of the
coefficients have been calculated by Pauling and Beach, loc. cit.
XIV47c]
VAN DER WAALS FORCES
387
47b. Van der Waals Forces for Helium. — In treating the
dipoledipole interaction of two helium atoms, the expression for
H ' consists of four terms like that of Equation 474, correspond
ing to taking the electrons in pairs (each pair consisting of an
electron on one atom and one on the other atom). The variation
function has the form
* = rh + Xhwu, r,)l.
Hass^ 1 has considered five variation functions of this form,
shown with their results in Table 472. The success of his similar
treatment of the polarizability of helium (function 6 of Table
293) makes it probable that the value lA13e 2 al/R« for W"
is not in error by more than a few per cent. Slater and Kirk
wood 1 obtained values 1.13, 1.78, and 1.59 for the coefficient
of — e 2 a\/R* by the use of variation functions based on their
helium atom functions mentioned in Section 29e. An approxi
mate discussion of dipolequadrupole and quadrupolequadrupole
interactions has been given by Margenau. 1
Table 472. — Variation Treatment of van der Waals Interaction of
Two Helium Atoms
r
f(nr 2 )
E  W°
i.
2.
3.
4.
5.
e Z't
e z>»
e z ''(\ +
e*'«(l +
CiU)
CiU)
A
A + B ri r 2
A + Br ir2 + Cr
A
A + Bt\ri
V 2
1.079e 2 a 5 /^
1.225
1.226
1.280
1.413
47c. The Estimation of van der Waals Forces from Molecular
Polarizabilities. — London 2 has suggested a rough method of
estimating the van der Waals forces between two atoms or mole
cules, based on the approximate secondorder perturbation
treatment of Section 27e. We obtain by this treatment (see
Sees. 27e and 29e) the expression
2ne 2 z 2
1 hoc. cit.
* F. London, Z. f. Phys. 63, 245 (1930).
(478)
388 MISCELLANEOUS APPLICATIONS [XIV48
for the polarizability of an atom or molecule, in which n is the
number of effective electrons, z* the average value of z 2 for these
electrons (z being the coordinate of the electron relative to
the nucleus in the field direction), and 7 the energy difference of
the normal state and the effective zero point for energy, about
equal in value to the first ionization energy. The van der Waals
interaction energy may be similarly written as
RVa + i B y ^ yj
which becomes on introduction of a A and a B
W = "2gr 77+T.' ( 47 " 10 >
or, in case the molecules are identical,
With a in units 10~ 24 cm 3 and I in volt electrons, this is
Dp 2 n*>
W" — — °
R« '
in which
D = 1.27a 2 /.
It must be realized that this is only a very rough approximation.
For hydrogen atoms it yields D = 7.65 (correct value 6.50) and
for helium 1.31 (correct value about 1.4).
For the further discussion of the validity of London's relation
between van der Waals forces and polarizabilities, and of other
applications of the relation, such as to the heats of sublimation
of molecular crystals and the unactivated adsorption of gases by
solids, the reader is referred to the original papers. 1
48. THE SYMMETRY PROPERTIES OF MOLECULAR WAVE
FUNCTIONS
In this section we shall discuss the symmetry properties of
molecular wave functions to the extent necessary for an under
1 F. London, loc. cit.; F. London and M. Polanyi, Z. f. phys. Chem, 11B,
222 (1930); M. Polanyi, Trans. Faraday Soc. 28, 316 (1932); J. E. Lbnnabd
Jones. ibid, 28, 333 (1932).
XIV481 PROPERTIES OF MOLECULAR WAVE FUNCTIONS 389
standing of the meaning and significance of the term symbols
used for diatomic molecules by the spectroscopist.
In Section 34 it was mentioned that the nuclear and electronic
parts of an approximate wave function for a molecule can be
separated by referring the electronic coordinates to axes deter
mined by the nuclear configuration. Let us now discuss this
choice of coordinates for a diatomic molecule in greater detail.
We first introduce the Cartesian coordinates X, Y, Z of the center
of mass of the two nuclei relative to axes fixed in space, and the
polar coordinates r, #, <p of nucleus A relative to a point midway
between nucleus A and nucleus B as origin, 1 also referred to axes
Fig. 481. — The relation between axes £, rj, f and X, Y, Z.
fixed in space, as indicated in Figure 481. We next introduce
the Cartesian coordinates £*, r) it £*» or the polar coordinates
r if #i, <pi of each of the electrons, measured with reference, not
to axes fixed in space, but instead to axes dependent on the
angular coordinates # and <p determining the orientation of the
nuclear axis. These axes, £, rj, f , are chosen in the following way.
f is taken along the nuclear axis OA (Fig. 481), and £ lies in the
XY plane, its sense being such that the Z axis lies between the
rj and f axes (£, r), f forming a lefthanded system, say). It is,
moreover, often convenient to refer the azimuthal angles of all
electrons but one to the azimuthal angle of this electron, using
the coordinates <p h (p2 — <pi, <Pz — <pi, • • • in place of <pi, <p 2 ,
<Pz, • • • .
1 It is convenient in this section to use these coordinates, which differ
slightly from those adopted in Chapter X.
390 MISCELLANEOUS APPLICATIONS [XIV48a
It has been shown 1 that these coordinates can be introduced
in the wave equation, and that the wave functions then assume
a simple form. We have discussed the wave function for the
nuclear motions in detail in Chapter X. The only part of the
electronic wave function which can be written down at once is
that dependent on <pi. Inasmuch as the potential energy of the
system is independent of <pi (as a result of our subterfuge of
measuring the <p's of the other electrons relative to <pi), <pi is a
cyclic coordinate, and occurs in the wave function only in the
factor e ±iA *i, in which A can assume the values 0, 1, 2, • • • .
The quantum number A thus determines the magnitude of the
component of electronic orbital angular momentum along
the line joining the nuclei. [A is somewhat analogous to the
component Ml of the resultant orbital angular momentum
(or azimuthal) quantum number L for atoms.] The value of A
is expressed by the principal character of a molecular term
symbol: 2 denoting A = 0; ft, A = ±1; A, A = ±2; etc. As
in the case of atomic terms, the multiplicity due to electron spin
is indicated by a superscript to the left, x 2 indicating a singlet,
2 2 a doublet, etc.
It may be mentioned that if we ignore the interactions of the
electronic and nuclear motions the wave functions corresponding
to A and —A correspond to identical energy values. This
degeneracy is removed by these interactions, however, which
lead to a small splitting of energy levels for A > 0, called Atype
doubling. 2 The correct wave functions are then the sum and
difference of those corresponding to A and —A.
In the following sections we shall discuss the characteristic
properties of diatomic molecules containing two identical nuclei
(symmetrical diatomic molecules).
48a. Even and Odd Electronic Wave Functions. Selection
Rules. — By the argument of Section 40e we have shown that the
transition probabilities for a diatomic molecule are determined
in the main by the electricmoment integrals over the electronic
parts of the wave functions, taken relative to the axes {, 17, f
determined by the positions of the nuclei. Let us now classify
the electronic wave functions of symmetrical diatomic molecules
1 F. Hund, Z. f. Phys. 42, 93 (1927); R. deL. Kronig, ibid. 46, 814; 50,
347 (1928); E. Wigner and E. E. Witmer, ibid. 51, 869 (1928).
s See, for example, J. H. Van Vleck, Phys. Rev. 33, 467 (1929).
XIV48b] PROPERTIES OF MOLECULAR WAVE FUNCTIONS 391
as even or odd, introducing the subscripts g (German gerade)
for even terms and u (ungerade) for odd terms in the term symbols
for identification. This classification depends on the behavior
of the electronic wave function with respect to the transformation
£1, Vi, f» — » —&, —Vi, —ft, that is, on inversion through the
origin, even functions remaining unchanged by this operation,
and odd functions changing sign. The argument of Section AQg
leads to the following selection rule: Transitions are allowed
only between even and odd levels (g —>u y u—tg).
(Although electronic wave functions for diatomic molecules
containing unlike nuclei cannot be rigorously classified as even
or odd, they often approach members of these classes rather
closely, and obey an approximate selection rule of the above
type.)
48b. The Nuclear Symmetry Character of the Electronic Wave
Function. — We are now in a position to discuss the nuclear
symmetry character of the electronic wave function for a diatomic
molecule in which the nuclei are identical. Interchanging the
two nuclei A and B converts # into t — & and <p into ir + <p;
these coordinates, however, do not occur in the electronic wave
function. The interchange of the nuclei also converts the
coordinates (■», v%j f* of each electron into — &, rn 9 — f», and hence
r if &i f <pi into r i} ir — & if tt — <pi [or <pi — <p x into — (^ — ^)].
In case that the electronic wave function is left unchanged by
this transformation, the electronic wave function is symmetric
in the nuclei; if the factor — 1 is introduced by the transformation,
the electronic wave function is antisymmetric in the nuclei.
The nuclear symmetry character of the electronic wave func
tion is represented in the term symbol by introducing the super
script + or — after taking cognizance of the presence of the
subscript g or u discussed in Section 48a, the combinations
g and u representing electronic wave functions symmetric in
the nuclei, and g and u those antisymmetric in the nuclei. Thus
we see that
2+ and 2" are S N
and
27 and 2+ are A*.
For A 9± there is little need to represent the symmetry char
acter in the term symbol, inasmuch as the S N and A N states
392
MISCELLANEOUS APPLICATIONS
[XIV48b
occur in pairs corresponding to nearly the same energy value
(Atype doubling), and in consequence the + and — superscripts
are usually omitted.
The states with superscript + are called positive states, and
those with superscript — negative states.
The principal use of the nuclear symmetry character is in
determining the allowed values of the rotational quantum
number X of the molecule. The complete wave functions for a
molecule (including the nuclearspin function) must be either
symmetric or antisymmetric in the nuclei, depending on the
nature of the nuclei involved. If the nuclei have no spins,
then the existent functions are of one or the other of the types
listed below.
I. Complete wave function aS^ :
+
9,
X even <
AX even  r ,
> u, K even
AX odd
AX odd
+ ^ , , AX even  ~ , ,
u, X odd < — > g, X odd
II. Complete wave function A N :
+ „ ,, AK even  r ,,
g } X odd < > u, K odd
AX odd
K even <
AX even
AX odd
+ g, K even
It is seen that in either case the transitions allowed by the selec
tion rule g ** u are such that AX is even for H > — or > +
transitions, and odd for  > + or > — transitions.
The selection rule AX = 0, ± 1 can be derived by the methods of
Chapter XI ; this becomeb AX = for positive «*> negative transi
tions, and AX = ± 1 for positive — * positive or negative — > nega
tive transitions.
In case that the nuclei possess spins, with spin quantum
number /, both types of functions and transitions occur (the two
not forming combinations), with the relative weights (/ + 1)// or
//(/ + 1), as discussed in Section 43f.
XIV48b] PROPERTIES OF MOLECULAR WAVE FUNCTIONS 393
Let us now consider a very simple example, in order to clarify
the question; namely, the case of a molecule possessing only one
electron, in the states represented by approximate wave functions
which can be built from the four orbitals u A = s, p s , p x , p v about
nucleus A, and four similar ones u B about nucleus B; s, p t , p x , p y
being real oneelectron wave functions such as given in Table
214 for the L shell. We can combine these into eight functions
of the form Sa + s B , s A — s B , etc. If the functions are referred
to parallel axes for the two atoms and taken as in Table 214
t
Fig. 482. — Positive and negative regions of wave functions b, p M , p x , and p v
for atoms A and B.
except for a factor —1 for p tB (introduced for convenience),
then they have the general nature shown in Figure 482, in which
the functions u A + u B are designated, the plus and minus signs
representing regions equivalent except for sign. From the
inspection of this figure and a similar one for u A — u B (in which
the signs are changed for u B ), it is seen that the eight functions
have the following symmetry character in the nuclei:
Function
8
V»
V*
Pv
UA + u B
ua — ub
S»
A»
S N
A N
A*
S"
s«
A*
394 MISCELLANEOUS APPLICATIONS [XIV48c
By the argument given above we know that four of these are
2 states, with A = 0, and four are II states. The II states are
those formed from p z and p v (which are the linear combinations
of the complex exponential functions p+ x and pi). The two
II states u A + u B are separated widely by the exchange integrals
from the two u A — u B , and the Atype doubling will cause a
further small separation of the nuclearsymmetric and nuclear
antisymmetric levels. The exchange terms similarly separate
the u A + u B s and p z functions from the u A — u B functions. The
best approximate wave functions would then be certain linear
combinations of the two nuclearsymmetric functions and also
of the two nuclearantisymmetric functions.
We can now write complete term symbols for the eight elec
tronic wave functions of our simple example, as follows:
« Pz Vx Vv
u A +u B *s+ 2 s; { 2 n+ 2 n}
u A u B '2+ *x: pn+ m}
The identification as even or odd is easily made by inspection of
Figure 482. The two 2 II W terms (one S N and one A N ) are placed
in brackets to show that they form a Atype doublet, as are the
two 2 U terms.
48c. Summary of Results Regarding Symmetrical Diatomic
Molecules. — The various symmetry properties which we have
considered are the following:
1. Even and odd electronic functions, indicated by subscripts
g and u (Sec. 48a). Selection rule: Transitions allowed only
between g and u.
2. The nuclear symmetry of the complete wave function
(including rotation of the molecule but not nuclear spin). Selec
tion rule: Symmetricantisymmetric transitions not allowed.
3. The nuclear symmetry of the electronic wave function,
represented by the superscripts + and — , g and u being S N ; g
and u, A N . Selection rule: AK = for positivenegative transi
tions, and AK = ± 1 f or positivepositive and negativenegative
transitions. (This is not independent of 1 and 2. In practice
1 and 3 are usually applied.)
We are now in a position to discuss the nature of the spectral
lines to be expected for a symmetrical diatomic molecule. We
XIV49] STATISTICAL QUANTUM MECHANICS 395
have not treated the spin moment vector of the electrons, which
combines with the angular momentum vectors A and K in various
ways to form resultants; the details of this can be found in the
treatises on molecular spectroscopy listed at the end of Chapter
X. Let us now for simplicity consider transitions among x 2
states, assuming that the nuclei have no spins, and that the
existent complete wave functions are symmetric in the nuclei
(as for helium). The allowed rotational states are then those
with K even for *2+ and ! 2~, and those with K odd for *2~
and l 2+ 9 and the transitions allowed by 1 and 3 are the following:
etc.
etc.
lv+
K =
2
4
\/ N 
i2+
K =
1
3
1 2 +
a
K =
2
4
1
1
i
12
K =
2
4
12"
K =
1
!
3
I
*2+
K =
l
3
IV
K =■
l
3
etc.
!2 K = 2 4
49. STATISTICAL QUANTUM MECHANICS. SYSTEMS IN
THERMODYNAMIC EQUILIBRIUM
The subject of statistical mechanics is a branch of mechanics
which has been found very useful in the discussion of the proper
ties of complicated systems, such as a gas. In the following
sections we shall give a brief discussion of the fundamental
theorem of statistical quantum mechanics (Sec. 49a), its applica
tion to a simple system (Sec. 496), the Boltzmann distribution
law (Sec. 49c), FermiDirac and BoseEinstein statistics (Sec.
49d), the rotational and vibrational energy of molecules (Sec. 49e),
and the dielectric constant of a diatomic dipole gas (Sec. 49/).
The discussion in these sections is mainly descriptive and
elementary ; we have made no effort to carry through the difficult
derivations or to enter into the refined arguments needed in a
396 MISCELLANEOUS APPLICATIONS [XIV49a
thorough and detailed treatment of the subject, but have
endeavored to present an understandable general survey.
49a. The Fundamental Theorem of Statistical Quantum
Mechanics. — Let us consider a large system with total energy
known to lie in the range W to W + AW. We inquire as to the
properties of this system. If we knew the wave function
representing the system, values of the dynamical quantities
corresponding to the properties of the system could be calculated
by the methods of Section 12d. In general, however, there will
be many stationary states of the system (especially if it be a very
complicated system, such as a sample of gas of measurable
volume) with energy values lying in the range W to W + AW y
and our knowledge of the state of the system may not allow us to
select one wave function alone as representing the system.
Moreover, it might be possible for us to find a set of approximate
wave functions for the system by ignoring weak interactions of
parts of the system with each other or of the system and its
environment; no one of these approximate wave functions
would represent the state of the system over any appreciable
period of time, and so we would not be justified in selecting any
one of them for use in calculating values of dynamical quantities.
Under these circumstances we might make calculations regard
ing the properties of the system for each of the wave functions
with energy between W and AW, and then average the various
calculations to obtain predictions regarding the average expected
behavior of the system. The important question immediately
arises as to what weights are to be assigned the various wave
functions in carrying out this averaging. The answer to this
question is given by the fundamental theorem of statistical
quantum mechanics, as follows: In calculating average values of
properties of a system with energy between W and AW y the same
weight is to be assigned to every accessible wave function with
energy in this range, in default of other information. (The wave
functions are of course to be normalized and mutually orthog
onal.) This theorem can be derived from the equations of
quantum mechanics (by methods such as the variation of con
stants, discussed in Chapter XI), with the aid of an additional
postulate, 1 which is the quantummechanical analogue of the
1 The postulate of randomness of phases. See, for example, W. Pauli,
"Probleme der modernen Physik," S. Hirzel, Leipzig, 1928.
XIV49b] STATISTICAL QUANTUM MECHANICS 397
ergodic hypothesis of classical statistical mechanics. We shall
not discuss this derivation.
The word accessible appears in the theorem for the following
reason. If a system is known to be in one state at a given
instant, and if it is known that it is impossible for any operative
perturbation to cause a transition to a certain other state, then
it is obviously wrong to include this latter state in the expression
for the average. We have already met such noncombining
states in our discussion of the symmetry of wave functions for
collections of identical particles (Sees. 296, 30a). It was shown
that if the system is known to be represented by a wave function
symmetrical in all the identical particles composing it, no
perturbation can cause it to change over to a state with an
antisymmetrical wave function. The nature of the wave
functions which actually occur is dependent upon the nature of
the system. If it is composed of electrons or protons, the wave
functions must be antisymmetric; if it is composed of hydrogen
atoms, thought of as entities, the wave functions must be sym
metric in these atoms; etc. Moreover, we may sometimes have
to take the passage of time into consideration in interpreting
the word accessible. Let us consider as our system a helium
atom, for example, which is known at the time t = to be in some
excited singlet state, the wave function being symmetric in the
positions of the electrons and antisymmetric in their spins.
Transitions to triplet states can occur only as a result of perturba
tions affecting the electron spins; and, since these perturbations
are very small, the probability of transition to all triplet states
in a short time will be very small. In predicting properties for
this system for a short period after the time t = 0, we would
accordingly be justified in considering only the singlet states as
accessible.
49b. A Simple Application. — In order to illustrate the use of
the fundamental theorem of statistical quantum mechanics, we
shall discuss a very simple problem in detail.
Let us consider a system composed of five harmonic oscillators,
all with the same characteristic frequency v } which are coupled
with one another by weak interactions. The set of product
wave functions ^(0)^(6)^(0)^(^)^(6) can be used to construct
approximate wave functions for the system by the use of the
method of variation of constants (Chap. XI). Here ¥(a), • • •
398
MISCELLANEOUS APPLICATIONS
[XIV49b
represent the harmonic oscillator wave functions (Sec. 11),
the letters a, b, c, d f e representing the coordinates of the five
oscillators. For each oscillator there is a set of functions ^n a (p)
corresponding to the values 0, 1, 2, • • • for the quantum
number n a . The total unperturbed energy of the system is
Wl = (n« + V 2 )hv + . . • + (n, + V 2 )hv = (n + %)hr, in
which n = n a + n b + n c + n d + n e .
The application of the variationofconstants treatment shows
that if the system at one time is known to have a total energy
value close to Wn>, where n f is a particular value of the quantum
number n, then the wave function at later times can be expressed
essentially as a combination of the product wave functions for
n = n', the wave functions for n ^ n ! making a negligible
contribution provided that the mutual interactions of the oscilla
tors are weak. Let us suppose that the system has an energy
value close to \2}^hv 9 that is, that n' is equal to 10. The product
wave functions corresponding to this value of n f are those
represented by the 1001 sets of values of the quantum numbers
tt 0> • • ' > n c given in Table 491.
Table 491. — Sets of Quantum Numbers for Five Coupled Harmonic
Oscillators with Total Quantum Number 10
n a
Tib
n c
rid
n e
10
.
Oetc.
* (5)
9
. 1
.
(20)
8
. 2
(20)
7
. 3
(20)
6
. 4
(20)
5
. 5
(10)
8
. 1
1
0
(30)
7
. 2
1
(60)
6
. 3
1
(60)
6
. 2
. 2
.
(30)
5
. 4
. 1
.
.
(60)
5
. 3
. 2
.
.
(60)
4
. 4
2
.
(30)
4
. 3
. 3
.
(30)
7
. 1
. 1
. 1
(20)
* The other sets indicated by "
. . . 10 . 0, and 0.
parentheses.
n a rib n c rid
n e
6.2.1.1
Oetc.
(60)
5.3.1.1
(60)
5.2.2.1
(60)
4 4.1.1
(30)
4.3.2.1
(120)
4.2.2.2
(20)
3.3.3.1
(20)
3.3.2.2
(30)
6.1.1.1
(5)
5.2.1.1
(20)
4.3.1.1
(20)
4.2 2.1
(30)
3.3 2.1
(30)
3.2.2.2
(20)
2 2.22
2
(1)
e . 10 . .
0, .
. 10 . . 0,
five, as shown
by the
number in
In case that the interactions between the oscillators are of a
general nature (the ab, ac f be, • • • interactions being different),
XIV49c] STATISTICAL QUANTUM MECHANICS 399
all of the product functions will be accessible, and the funda
mental theorem then requires that over a long period of time the
1001 product functions will contribute equally to the wave
function of the system. In calculating the contribution of
oscillator a, for example, to the properties of the system, we
would calculate the properties of oscillator a in the states n a —
[using the wave function ^o(a)L n « = 1, * * ' , n a = 10, and then
0.4
0.3
Pn 02
0.1
0.0
•23456789 10 II
n a >
Fig. 491. — The probability values P n for systempart a in a system of five
coupled harmonic oscillators with total quantum number n = 10 (closed circles),
and values calculated by the Boltzmann distribution law (open circles).
average them, using as weights the numbers of times that
n a = 0, 1, 2, • • • , 10 occur in Table 491. These weights are
given in Table 492. The numbers obtained by dividing by
the total (1001) can be described as the probabilities that oscilla
tor a (or b, Cj • • • ) be in the states n a = 0, 1, 2, • • • , 10.
These probability values are represented graphically in Figure
491.
49c. The Boltzmann Distribution Law. — We have been dis
cussing a system composed of a small number (five) of weakly
interacting parts. A similar discussion (which we shall not
give because it is necessarily rather involved) of a system com
posed of an extremely large number of »veakly interacting parts
can be carried through, leading to a general expression for the
probability of distribution of any one of the parts among its
400
MISCELLANEOUS APPLICATIONS
[XIV49C
Tablk 492. — Weights fob States of Individual Oscillators in
Coupled System
n , etc.
Weight
Probability P„ a
286
0.286
1
220
.220
2
165
.165
3
120
.120
4
84
.084
5
56
.056
6
35
.035
7
20
.020
8
10
.010
9
4
.004
10
1
.001
Total
1001
1.001
stationary states. 1 The result of the treatment is the Boltzmann
distribution law in its quantummechanical form:
// all the product wave functions ^ (a) V(b) • • • of a system
composed of a very large number of weakly interacting parts a,
by * • • are accessible, then the probability of distribution of one
of the parts, say a, among its states, represented by the quantum
number n a , is given by the equation
P na  Ae""*?, (491)
in which W n is the energy of the part a in its various states and the
constant A has such a value as to make
X*\  i
(492)
There is considered to be one state for every independent wave
function ^(a). The exponential factor, called the Boltzmann
exponential factor, is the same as in the classical Boltzmann
distribution law, which differs from Equation 491 only in the
way the state of the system part is described. The constant k
is the Boltzmann constant, with the value 1.3709 X 10 16 erg
deg 1 . The absolute temperature T occurring in Equation 491
1 That is, among the stationary states for this part of the system when
isolated from the other parts.
XIV49C] STATISTICAL QUANTUM MECHANICS 401
is introduced in the derivation of this equation by methods
closely similar to those of classical statistical mechanics.
Some indication of the reasonableness of this equation is given
by comparing it with the results of our discussion of the system of
five coupled harmonic oscillators. The open circles in Figure
491 represent values of P n<j calculated by Equation 491, with
kT placed equal to %hv (this leading approximately to the average
value Yilftv for W n<x , as assumed in the earlier discussion). It is
seen that there is general agreement, the discrepancies arising
from the fact that the number of parts of the system (five) is
small (rather than very large, as required in order that the
Boltzmann distribution law be applicable).
In Equation 491 each wave function is represented separately.
It is often convenient to group together all wave functions
corresponding to the same energy, and to write
Pi = Ap x e kT , (493)
in which pi is the degree of degeneracy or a priori probability or
quantum weight of the energy level Wi.
In case that the wave functions for the part of the system
under consideration are very numerous and correspond to energy
values lying very close together, it is convenient to rewrite the
distribution law in terms of P(W), such that P(W)dW is the
probability that the energy of the system part lie between W
and W + dW, in the form
_w_
P{W) = Ap(W)e kT , (494)
in which p(W)dW is the number of wave functions for the system
part in the energy range W to W + dW.
As an illustration of the use of Equation 49^i let us consider
the distribution in translational energy of the molecules of a
gas (the entire gas being the system and the molecules the system
parts) such that all product wave functions are accessible. 1
It is found (by the use of the results of Section 14, for example)
that p(W) is given by the equation
vW) = ^!!p, (495)
1 We shall see in the next section that actual gases are not of this type.
402 MISCELLANEOUS APPLICATIONS [XIV49d
in which V is the volume of the box containing the gas and m
is the mass of a molecule. The Maxwell distribution law for
velocities is obtained by substituting this in Equation 494 and
replacing W by y^mv 2 , v being the velocity of the molecule.
Problem 491. Derive Equation 495 with the use of the results of
Section 14. By equating W to the kinetic energy ]^mv 2 (v being the
velocity), derive the Maxwell distribution law for velocities, and from it
calculate expressions for the mean velocity and rootmeansquare velocity
of gas molecules.
It will be shown in the following section that the Boltzmann
distribution law is usually not strictly applicable in discussing
the translational motion of molecules.
49d. FermiDirac and BoseEinstein Statistics. — As stated in
the foregoing section, the Boltzmann distribution law is applicable
to the parts of a system for which all product wave functions are
accessible. The parts of such a system are said to conform to
Boltzmann statistics. Very often, however, we encounter systems
for which not all product wave functions are accessible. We
have seen before (Sec. 29, etc.) that the wave functions for a
system of identical particles can be grouped into noncombining
sets of different symmetry character, one set being completely
symmetric in the coordinates of the particles, one completely
antisymmetric, and the others of intermediate symmetry char
acter. Only the wave functions of one symmetry character are
accessible to a given system of identical particles.
Thus our simple system of five harmonic oscillators would be
restricted to wave functions of one symmetry character if the
interactions ab y ac } 6c, • • • were equivalent, that is, if the
oscillators were identical. 1 It was to avoid this that we made
the explicit assumption of nonequivalence of the interactions in
Section 4%. The accessible wave functions for five identical
oscillators would be the completely symmetric ones, the com
1 In order for the oscillators to behave identically with respect to external
perturbations as well as mutual interactions they would have to occupy the
same position in space; that is, to oscillate about the same point. A
system such as a crystal is often treated approximately as a set of coupled
harmonic oscillators (the atoms oscillating about their equilibrium posi
tions). The Boltzmann statistics would be used for this set of oscillators,
inasmuch as the interactions depend on the positions of the oscillators in
space in such a way as to make them nonidentical.
XIV49dJ STATISTICAL QUANTUM MECHANICS 403
pletely antisymmetric ones, or those with the various inter
mediate symmetry characters. It is only the two extreme types
which have been observed in nature. There are 30 completely
symmetric wave functions for n = 10; they are formed from the
successive sets in Table 491 by addition, the first being
~{ (10.0.0.0.0) + (0.10.0.0.0) + (0.0.10.0.0) + (0.0.0.10.0) +
(0.0.0.0.10))
and the last being (2.2.2.2.2). From these we can obtain
weights for the successive values, similar to those given in
Table 492; these weights will not be identical with those of the
table, however, and so will correspond to a new statistics. This
is very clearly seen for the case that only the completely anti
symmetric wave functions are accessible. The only wave
function with n = 10 which is completely antisymmetric is that
formed by suitable linear combination of the 120 product func
tions (4.3.2.1.0), etc., marked A in Table 491 (the other functions
violate Pauli's principle, the quantum numbers not being all
different). Hence even at tjie lowest temperatures only one of
the five oscillators could occupy the lowest vibrational state,
whereas the Boltzmann distribution law would in the limit
T — » place all five in this state.
If only the completely antisymmetric wave functions are accessible
to a system composed of a large number of weakly interacting parts,
the system parts conform to the FermiDirac statistics; 1 if only the
completely symmetric wave functions are accessible, they conform
to the BoseEinstein statistics. 2
The FermiDirac distribution law in the forms analogous to
Equations 491, 493, and 494 is
P n = —^ > (496)
Ae kT +N
1 E. Fermi, Z. f. Phys. 36, 902 (1926); P. A. M. Dirac, Proc. Roy. Soc.
A112, 661 (1926). This statistics was first developed by Fermi, on the basis
of the Pauli exclusion principle, and was discovered independently by Dirac,
using antisymmetric wave functions.
2 S. N. Bosb, Z.f. Phys. 26, 178 (1924); A. Einstein, Sitzber. Preuss. Akad.
Wiss. p. 261, 1924; p. 3, 1925. Bose developed this statistics to obtain a
formal treatment of a photon gas, and Einstein extended it to the case of
material gases.
404 MISCELLANEOUS APPLICATIONS [XIV49d
Pi = —^ > (497)
Ae kT +N
and
P(W) = y°j (498)
in each of which the constant A has such a value as to make the
sum or integral of P equal to unity. Here N is the number of
identical system parts in which the accessible wave functions
are antisymmetric.
Problem 492. Show that at very low temperatures the FermiDirac
distribution law places one system part in each of the N lowest states.
The FermiDirac distribution law for the kinetic energy of the
particles of a gas would be obtained by replacing p(W) by the
expression of Equation 495 for point particles (without spin)
or molecules all of which are in the same nondegenerate state
(aside from translation), or by this expression multiplied by the
appropriate degeneracy factor, which is 2 for electrons or protons
(with spin quantum number Yi) ) or in general 27 + 1 for spin
quantum number I. This law can be used, for example, in dis
cussing the behavior of a gas of electrons. The principal
application which has been made of it is in the theory of metals, *
a metal being considered as a first approximation as a gas of
electrons in a volume equal to the volume of the metal.
Problem 493. (a) Evaluate the average kinetic energy of the valence
electrons (ignoring the K electrons and the nuclei) in a crystal of lithium
metal at 0°A, and discuss the distribution of energy, (b) Calculate the
number of electrons at 298 C A with kinetic energy 0.10 v.e. greater than the
maximum for 0°A. The density of lithium is 0.53 g./cm 3 .
The BoseEinstein distribution law in the forms analogous to
Equations 496, 497, and 498 is
P« = — ^ 1 (499)
Ae kT  N
* W. Patjli, Z. /. Phys. 41, 81 (1927); A. Sommerfeld, Z. /. Phys. 47, 1,
43 (1928); etc. Review articles have been published by K. K. Darrow,
Rev. Mod. Phys. 1, 90 (1929); J. C. Slater, Rev. Mod. Phys. 6, 209 (1934);
etc.
XIV49e] STATISTICAL QUANTUM MECHANICS 405
Pi = — ^p > (4&10)
Ae**  N
and
P(W) = ?£P—> (4911)
Ae* r  N
in which the symbols retain their former significance. The Bose
Einstein statistics is to be used for photons, 1 deuterons, helium
atoms, hydrogen molecules, etc.
For many systems to which FermiDirac or BoseEinstein
statistics is to be applied the term ±N is negligible compared to
J?
Ae kT , and the appropriate equations are very closely approxi
mated by the corresponding Boltzmann equations. Thus helium
gas under ordinary conditions shows no deviations from the
perfect gas laws (Boltzmann statistics) which can be attributed
to the operation of BoseEinstein statistics. At very low
temperatures and very high pressures, deviations due to this
cause should occur, however; this degeneration 2 has not been
definitely shown to occur for material gases by experiment, 3
the principal difficulty being that real gases elude investigation
under extreme conditions by condensing to a liquid or solid
phase.
49e. The Rotational and Vibrational Energy of Molecules. —
In the statistical discussion of any gas containing identical
molecules, cognizance must be taken of the type of statistics
applicable. Often, however, we are not primarily interested in
the translational motion of the molecules but only in their dis
tribution among various rotational, vibrational, and electronic
states. This distribution can usually be calculated by the use
of the Boltzmann distribution law, the effect of the symmetry
character being ordinarily negligible (except in so far as the sym
1 With appropriate modifications to take account of the vanishing rest
mass of photons.
* The word degeneracy is used in this sense (distinct from that of Section
14), the electrons in a metal being described as constituting a degenerate
electron gas.
* G. E. Uhlenbeck and L. Gboppeb, Phys. Rev. 41, 79 (1932), and refer
ences there quoted.
406 MISCELLANEOUS APPLICATIONS [XIV49e
metry character relative to identical particles in the same
molecule determines the allowed wave functions for the molecule).
In case that the energy of a molecule can be represented as the
sum of several terms (such as rotational, vibrational, electronic,
and translational energy), the Boltzmann factor can be written
as the product of individual Boltzmann factors, and the con
tributions of the various energy terms to the total energy of the
system in thermodynamic equilibrium and to the heat capacity,
entropy, and other properties can be calculated separately. To
illustrate this we shall discuss the contributions of rotational and
vibrational motion to the energy content, heat capacity, and
entropy of hydrogen chloride gas.
As shown in Chapter X, the energy of a hydrogen chloride
molecule in its normal electronic state can be approximately
represented as
W VtK = (v + V 2 )hv + K(K + 1)J^, (4912)
in which v is the vibrational frequency, I the moment of inertia
of the molecule, and v and K the vibrational and rotational
quantum numbers, with allowed values v = 0, 1, 2, • • • and
K = 0, 1, 2 • • • . At all but very high temperatures the
Boltzmann factor for excited electronic states is very small, so
that only the normal electronic state need be considered. Using
Equation 493, we write for the probability that a molecule be
in the state v,K the expression
PvK = PvPk
(4913)
in which
(»+K)*»
P, = Be kT ,
(4914)
and
K(K+l)h*
Pk
= C(2K + l)e »"»* ,
(4915)
2K + 1 being the quantum weight of the Kth rotational state.
B and C have values such that
00 00
%P V = 1 and 2 P * = L
»0 JK>0
It is seen that the average rotational and vibrational energy
per molecule can hence be written as
XTV49e] STATISTICAL QUANTUM MECHANICS
407
oo oe
W = 22 P ' Pjt { (1 ' + l/i)hv + K{K + 1} 8^?}'
or, since the summation over K can be at once carried out for the
first term (to give the factor 1) and that over v for the second
term,
with
w vlhT . = Jo* + y 2 )hvP,
and
that is, the average energy is separable into two parts in the
same way as the energy W v ,k (Eq. 4912). By introducing the
variables 1
hv
kT
h 2
(4916)
8ir 2 IkT
these parts can be written as
00
%(v + y 2 )xe~^W*
JFvibr. = kT
» =
and
J K(K + 1)(2K + l)<re*<*+i>r
(4917)
W K t, = kT
K~0
%(2K + 1)€
(4918)
JT(JMl)*
25T =
l The symbol <r is conventionally used in this way as well as for the
quantity A 2 /8r 2 /, as in Section 35.
408 MISCELLANEOUS APPLICATIONS [XIV4W
the sums in the denominators corresponding to the factors JB
and C of Equations 4914 and 4915. Expressions for the
vibrational and rotational heat capacity C V ibr. and C ro t. can be
obtained by differentiating with respect to T y and the contribu
tions of vibration and rotation to the entropy can then be
obtained as S vib , = f ^'dT and S rot . = f ^fdT.
Problem 494. Considering only the first two or three excited states,
calculate the molal vibrational energy, heat capacity, and entropy of
hydrogen chloride at 25°C, using the vibrational wave number v —
2990 cm" 1 .
Problem 496. By replacing the sums by integrals, show that the
expressions 4917 and 4918 approach the classical value kT for large T.
Problem 496. Calculate the rotational energy curve (as a function of T)
for hydrogen chloride at temperatures at which it begins to deviate from
zero. The internuclear distance is 1.27 A.
The treatment of ortho and para hydrogen, mentioned in
Section 43/, differs from that of hydrogen chloride only in the
choice of accessible rotational wave functions. For para hydro
gen K can assume only the values 0, 2, 4, • • • , the quantum
weight being 2K + 1. For ortho hydrogen K can have the
values 1, 3, 5, • • • , with quantum weight S(2K + 1), the
factor 3 being due to the triplet nuclearspin functions. Ordinary
hydrogen is to be treated as a mixture of onequarter para and
threequarters ortho hydrogen, inasmuch as only the states with
K even are to be considered as accessible to the para molecules,
and those with K odd to the ortho molecules. In the presence
of a catalyst, however, all states become accessible, and the gas
is to be treated as consisting of molecules of a single species.
Problem 497. Discuss the thermodynamic properties (in their depend
ence on rotation) of the types of hydrogen mentioned above.
Problem 498. Similarly treat deuterium and protiumdeuterium
molecules (see footnote, Sec. 43/).
49f. The Dielectric Constant of a Diatomic Dipole Gas. —
Under the influence of an electric field, a gas whose molecules
have a permanent electric moment and in addition can have a
further moment induced in them by electronic polarization
becomes polarized in the direction of the field, the polarization
per unit volume being
XIV49f] STATISTICAL QUANTUM MECHANICS 409
P = g jqrg*' = N* + NaF > (4919)
in which € is the dielectric constant of the gas, F the strength of
the applied field (assumed to be parallel to the z axis), N the
number of molecules in unit volume, and a the polarizability of
the molecule. JT Z represents the average value of JT t for all
molecules in the gas, JT Z being the average value of the z compo
nent of the permanent electric moment /x of a molecule in a given
state of motion. It was shown by Debye 1 that according to
classical theory ]T Z has the value
JT.  ^ (4920)
We shall now show that for the special case of a diatomic dipole
gas, such as hydrogen chloride, the same expression is given
by quantum mechanics.
Let us consider that the change of the permanent moment
fjL with change in the vibrational quantum number v can be
neglected. ]T Z is then given by the equation
f z = *2fKuWu(KM), (4921)
K,M
in which 2
Pkm = Ae KiK + l) *, (4922)
with cr = h 2 /8ir 2 IkT> as in Equation 4916. Our first task is
hence to evaluate 'JT Z {KM) 1 which is the average value of
fX z = n cos #
for a molecule in the rotational state described by the quantum
numbers K and M, & being the angle between the moment /x of
the molecule (that is, the nuclear axis) and the z axis.
The value of JT Z (KM) is given by the integral
Wz{KM) = S+* M p cos tty*A,dr, (4923)
in which ^ K m is the firstorder perturbed wave function for the
molecule in the electric field. It is found, on application of the
1 P. Debye, Phys. Z. 13, 97 (1912).
* It is assumed at this point that the energy of interaction of the molecule
and the field can be neglected in the exponent of the Boltzmann factor.
An investigation shows that this assumption is valid
410 MISCELLANEOUS APPLICATIONS [XIV49f
usual methods of Chapters VI and VII, the perturbation function
being
H' = fjiFcoad, (4924)
that jr x (KM) has the value
r KM \  8t2I » 2F & M2  g(g + l l \ awn
H^am)  h2 (2X^ 1)K(K + 1)(2K + S) W M >
(see Prob. 499).
Inasmuch as P K m is independent of the quantum number M ,
to the degree of approximation of our treatment, we can at once
calculate the average value of jT z (KM) for all states with the same
value of K y by summing jr,(KM) for M = —K,K + 1, • • • ,
+K, and dividing by 2K + 1. The only part of 4925 which
+ K
involves any difficulty is that in M 2 . The value of V M 2
MK
is HK(K + 1)(2K + 1); using this, we see that
+ K
UK) = gAj ^ ^ M ) = °> * > ° ( 49 ~ 26 )
MK
Thus we have obtained the interesting result that the only
rotational state which contributes to the polarization is that with
K = 0. The value of JT t for this state is seen from Equation
4925 to be
WM = ^g?> (4927)
and fit hence is given by the equation
= . gP __^__ ., (49 _ 28)
c 2 (2K + 1)€
2T0
in which the sum in the denominator corresponds to the constant
A of Equation 4922. For small values of <r (such as occur in
actual experiments) this reduces to
= _ n 2 F
M * " ZkT'
(4929)
which is identical with the classical expression 4920. On
introduction in Equation 4919, this gives the equation
XIV49fl STATISTICAL QUANTUM MECHANICS 411
Problem 499. Using the surfaceharmonic wave functions mentioned
in the footnote at the end of Section 35c, derive Equation 4925, applying
either the ordinary secondorder perturbation theory or the method of
Section 27a.
Problem 4910. Discuss the approximation to Equation 4928 provided
by 4929 for hydrogen" chloride molecules (m ~ 1.03 X 10"" 18 e.s.u.) in a
field of 1000 volts per centimeter.
It can be shown 1 that Equation 4930 is not restricted to dia
tomic molecules in its application, but is valid in general, except
for a few special cases (as, for example, for a molecule with
electric moment largely dependent on the vibrational state, or
on the state of rotation of one part of the molecule about a single
bond, etc.). With the use of this equation the electric moments
of molecules can be determined from measurements on the
temperature coefficient of the dielectric constants of gases and
dilute solutions and in other ways. This has been done for a very
large number of substances, with many interesting structural
conclusions. An illustration is the question of which of the two
isomers of dichlorethylene is the cis and which the trans form,
i.e., which compound is to be assigned to each of the formulas
shown below:
H H
\ /
C=C
/ \
CI CI
cis form
The trans form is symmetrical and therefore is expected to have
zero electric moment. It is found experimentally that the
compound which the chemists had previously selected as the
trans form does in fact have zero moment, whereas the cis form
has a moment of about 1.74 X 10~ 18 e.s.u. (The unit 10 18 e.s.u.
is sometimes called a Debye unit.) Strong evidence for the
plane structure of benzene is also provided by electricmoment
data, and many other problems of interest to chemists have been
attacked in this way.
1 See the references at the end of the section, in particular Van Vleck.
H
CI
\
/
C=
=C
/
\
CI
H
trans form
412 MISCELLANEOUS APPLICATIONS [XIV50
An equation which is very closely related to Equation 4930 is
also applicable to the magnetic susceptibility of substances;
indeed, this equation was first derived (by Langevin 1 in 1905)
for the magnetic case. The temperaturedependent term in this
case corresponds to paramagnetism, fx representing the magnetic
moment of the molecule; and the other term, which in the mag
netic case is negative, corresponds to diamagnetism. For
discussions of the origin of diamagnetism, the composition of the
resultant magnetic moment /x from the spin and orbital moments
of electrons, etc., the reader is referred to the references men
tioned below.
References on Magnetic and Electric Moments
J. H. Van Vleck: "The Theory of Electric and Magnetic Suscepti
bilities," Oxford University Press, 1932.
C. P. Smyth: "Dielectric Constant and Molecular Structure, ,, Chemical
Catalog Company, Inc., New York, 1931.
P. Debye: "Polar Molecules," Chemical Catalog Company, Inc., New
York, 1929.
E. C. Stoner: "Magnetism and Atomic Structure," E. P. Dutton &
Co., Inc., New York, 1926.
The most extensive table of values of dipole moments available at present
is that given in an Appendix of the Transactions of the Faraday Society, 1934.
General References on Statistical Mechanics
R. C. Tolman: "Statistical Mechanics with Applications to Physics and
Chemistry," Chemical Catalog Company, Inc., New York, 1927.
R. H. Fowler: "Statistical Mechanics," Cambridge University Press,
1929.
L. Brillotjin: "Les Statistiques Quantiques," Les presses universitaires de
France, Paris, 1930.
K. K. Darrow: Rev. Mod. Phys. 1, 90 (1929).
R. H. Fowler and T. E. Sterne: Rev. Mod. Phys. 4, 635 (1932).
50. THE ENERGY OF ACTIVATION OF CHEMICAL REACTIONS
A simple interpretation of the activation energy E of a chemical
reaction such as
A + BC > AB + C (501)
is provided by the assumption that the molecule BC in its normal
electronic state is not able to react with the atom A, and that
reaction occurs only between A and an electronically excited
molecule BC*, E being then the energy difference of the normal
1 P. Langbvin, J. de phys. 4, 678 (1905).
XIV60J ENERGY OF ACTIVATION OF CHEMICAL REACTIONS 413
and the excited molecule. A reasonable alternative to this
was given in 1928 by London, 1 who suggested that such a reaction
might take place without any change in the electronic state of
the system (other than that accompanying the change in the
internuclear distances corresponding to the reaction 501, as
discussed in Section 34). The heat of activation would then be
obtained in the following way. We consider the electronic
energy TF (£) for the normal electronic state of the system as a
'AB
'BC
Fig. 501. — The electronic energy surface (showing contour lines with
increasing energy 1, 2, 3, etc.) for a system of three atoms arranged iinearly, as a
function of the internuclear distances tab and tbc.
function of the nuclear coordinates f . W (%) will have one value
for the nuclear configuration in which nuclei B and C are close
together, as in the normal molecule BC, and A is far removed,
and another value for the AB + C nuclear configuration. (The
difference of these, corrected for the energy of oscillation and
rotation of the molecules, is the energy change during the
reaction.) Now in order to change from one extreme configura
tion to the other, the nuclei must pass through intermediate
configurations, as atom A approaches B and C recedes from it,
and the electronic energy Wo(£) would change with change in
1 F. London in the Sommerfeld Festschrift, "Probleme der modernen
Phyaik/'p. 104, S. Hirzel, Leipzig, 1928.
414 MISCELLANEOUS APPLICATIONS [XIV50
configuration, perhaps as shown in Figure 501. The change
from A + BC, represented by the configuration point P, to
AB + C, represented by the configuration point P", could take
place most easily along the path shown by the dotted line.
We have seen in Section 34 that the electronic energy can be
treated as a potential function for the nuclei ; it is evident that in
order for reaction to take place the nuclei must possess initially
enough kinetic energy to carry them over the high point P' of
the saddle of the potential function of Figure 501. The energy
difference Wo(P') — Wo(P) t after correction for zeropoint
oscillational energy, etc., would be interpreted as the activation
energy E.
No thoroughly satisfactory calculation of activation energies in
this way has yet been made. The methods of treatment dis
cussed for the hydrogen molecule in Section 43, in particular the
method of James and Coolidge, could of course be extended to a
system of three protons and three electrons to provide a satis
factory treatment of the reaction H + H 2 — > H2 + H. This
calculation would be difficult and laborious, however, and has
not been carried out. Several rough calculations, providing
values of E for comparison with the experimental value 1 of
about 6 kcal/mole (from the orthopara hydrogen conversion),
have been made. In Section 46c? we have seen that at large
distances the interaction of a hydrogen atom A and a hydrogen
molecule BC is given approximately by the expression
 y 2 (abc \H\ hoc)  y 2 (abc \H\ cba),
the first term corresponding to repulsion of A by B and the second
to repulsion of A by C. It is reasonable then that the easiest
path for the reaction would correspond to a linear arrangement
ABC, the repulsion of A and C then being a minimum for given
values of tab and r B c Eyring and Polanyi 2 calculated energy
surfaces for linear configurations by neglecting higher exchange
integrals and making other simplifying assumptions, the values
1 A. Farkas, Z. f. phys. Chem. BIO, 419 (1930); P. Harteck and K. H.
Gbib, ibid. B15, 116 (1931).
2 H. Eyring and M. Polanyi, Naturwissenschaften 18, 914 (1930); Z. f.
phy*. Chem. B12, 279 (1931); H. Eyring, Naturwissenschaften 18, 915
(1930), /. Am. Chem. Soc. 63, 2537 (1931); H. Pelzer and E. Wigner,
Z.f. phys. Chem. B15, 445 (1932).
XIV60] ENERGY OF ACTIVATION OF CHEMICAL REACTIONS 415
of the Coulomb and single exchange integrals being taken from
the simple HeitlerLondonSugiura treatment of the hydrogen
molecule or estimated from the empirical potential function for
this molecule. These approximate treatments led to values
in the neighborhood of 10 to 15 kcal for the activation energy.
Coolidge and James 1 have recently pointed out that the approxi
mate agreement with experiment depends on the cancellation of
large errors arising from the various approximations.
The similar discussion of the activation energies of a number of
more complicated reactions has been given by Eyring and
collaborators. 2
1 A. S. Coolidge and H. M. James, /. Chem. Phys. 2, 811 (1934).
2 H. Eyring, /. Am. Chem. Soc. 53, 2537 (1931); G. E. Kimball and
H. Eyring, ibid. 64, 3876 (1932); A. Sherman and H. Eyring, ibid. 64,
2661 (1932); R. S. Bear and H. Eyring, ibid. 56, 2020 (1934); H. Eyring,
A. Sherman, and G. E. Kimball, /. Chem. Phys. 1, 586 (1933); A. Sherman,
C. E. Sun, and H. Eyring, ibid. 3, 49 (1935).
CHAPTER XV
GENERAL THEORY OF QUANTUM MECHANICS
The branch of quantum mechanics to which we have devoted
our attention in the preceding chapters, based on the Schrodinger
wave equation, can be applied in the discussion of most questions
which arise in physics and chemistry. It is sometimes conven
ient, however, to use somewhat different mathematical methods ;
and, moreover, it has been found that a thoroughly satisfactory
general theory of quantum mechanics and its physical inter
pretation require that a considerable extension of the simple
theory be made. In the following sections we shall give a brief
discussion of matrix mechanics (Sec. 51), the properties of angular
momentum (Sec. 52), the uncertainty principle (Sec. 53), and
transformation theory (Sec. 54).
61. MATRIX MECHANICS
In the first paper written on the quantum mechanics 1 Heisen
berg formulated and successfully attacked the problem of calcu
lating values of the frequencies and intensities of the spectral
lines which a system could emit or absorb; that is, of the energy
levels and the electricmoment integrals which we have been
discussing. He did not use wave functions and wave equations,
however, but instead developed a formal mathematical method
for calculating values of these quantities. The mathematical
method is one with which most chemists and physicists are not
familiar (or were not, ten years ago), some of the operations
involved being surprisingly different from those of ordinary
algebra. Heisenberg invented the new type of algebra as he
needed it; it was immediately pointed out by Born and Jordan, 2
however, that in his new quantum mechanics Heisenberg was
1 W. Heisenberg, Z. /. Phya. 33, 879 (1925).
1 M. Born and P. Jordan, ibid. 34, 858 (1925).
416
XV51a] MATRIX MECHANICS 417
making use of quantities called matrices which had already been
discussed by mathematicians, and that his newly invented
operations were those of matrix algebra. The Heisenberg
quantum mechanics, usually called matrix mechanics, was rapidly
developed 1 and applied to various problems.
When Schrodinger discovered his wave mechanics the question
arose as to the relation between it and matrix mechanics. The
answer was soon given by Schrodinger 2 and Eckart, 3 who showed
independently that the two are mathematically equivalent.
The arguments used by Heisenberg in formulating his quantum
mechanics are extremely interesting. We shall not present
them, however, nor enter into an extensive discussion of matrix
mechanics, but shall give in the following sections a brief treat
ment of matrices, matrix algebra, the relation of matrices to wave
functions, and a few applications of matrix methods to quantum
mechanical problems.
51a. Matrices and Their Relation to Wave Functions. The
Rules of Matrix Algebra. — Let us consider a set of orthogonal
wave functions 4 ^ , ^1, • • • , ^n, • • • and a dynamical quantity
f(qi, pi) 9 the corresponding operator 6 being f ol
=f (*>5n£)
In the foregoing chapters we have often made use of integrals
such as
fmn = fV&UVndr; (511)
for example, we have given f nn the physical interpretation of the
average value of the dynamical quantity / when the system is in
the nth stationary state. Let us now arrange the numbers
f mn (the values of the integrals) in a square array ordered accord
ing to m and n, as follows :
1 M. Born, W. Heisenberg, and P. Jordan, Z. f. Phys. 36, 557 (1926);
P. A. M. Dirac, Proc. Roy. Soc. A109, 642 (1925).
* E. Schrodinger, Ann. d. Phys. 79, 734 (1926).
• C. Eckart, Phys. Rev. 28, 711 (1926).
4 These functions include the time factor; a similar discussion can be made
with use of the functions ^ , 4n 9 • • • not including the time.
8 In this chapter we shall use the symbol / op . to represent the operator
corresponding to the dynamical function /. The subscript "op." was not
used in the earlier chapters because there was no danger of confusion attend
ing its omission. See Sees. 10, 12.
if 00
At
/02
/03 • .  1
Ao
Ai
/«
/l3 ...
/20
At
jii
/23 ...
J3C
1. . .
At
fzi
/33 ...
,
418 GENERAL THEORY OF QUANTUM MECHANICS [XV61a
f = (fmn) =
This array we may represent by the symbol f or (/ mn ). We
enclose it in parentheses to distinguish it from a determinant,
with which it should not be confused.
We can construct similar arrays g, h, etc. for other dynamical
quantities.
It is found that the symbols f, g, h, etc. representing such
arrays can be manipulated by an algebra closely related to ordi
nary algebra, differing from it mainly in the process of multiplica
tion. The rules of this algebra can be easily derived from the
properties of wave functions, which we already know.
It must be borne in mind that the symbol f does not represent
a single number. (In particular the array f must not be confused
with a determinant, which is equal to a single number. There is,
to be sure, a determinant corresponding to each array, namely,
the determinant whose elements are those of the array. We have
set up such determinants in the secular equations of the preceding
chapters.) The symbol f instead represents many numbers —
as many as there are elements in the array. The sign of equality
in the equation f = g means that every element in the array f is
equal to the corresponding element in the array g.
Now let us derive some rules of the new algebra. For example,
the sum of two such arrays is an array each of whose elements
is the sum of the corresponding elements of the two arrays; that
is,
//00 + (700 /oi + 001 /02 +^02 • * *\
f + g = ( /10 + 0io /n + 0ii /12 + 012 • • l (512)
It is seen that the arrays add in the same way as ordinary
algebraic quantities, with f + g = g + f . Addition is com
mutative.
On the other hand, multiplication is not commutative: the product
fg is not necessarily equal to the product gf. Let us evaluate the
mnth element of the array fg. It is
\fg] mn = J^lfop.0op.^ndT.
XV51a] MATRIX MECHANICS 419
Now we can express the quantity g OD & n in terms of the functions
SFa; with constant coefficients (Sec. 22), obtaining
k
That the coefficients are the quantities g kn is seen on multiplying
by ¥£ and integrating. Introducing this in the integral for
{fg}mn we obtain
{fg}mn = %S*ZU.*kdTg kn ;
k
since j^^f OJ> ^ k dr is equal to / m *, this becomes
\fg\mn = ^fmkgicn. (51"3)
ft
This is the rule for calculating the elements of the array obtained
on multiplying two arrays.
We may continue to develop the algebra of our arrays in this
way; or we can instead make use of work already done by mathe
maticians. The arrays which we have been discussing are called
matrices y and their properties have been thoroughly investigated
by mathematicians, who have developed an extensive matrix
algebra, 1 some parts of which we have just derived.
Problem 511. Show that the laws of ordinary algebra hold for the
addition and subtraction of matrices and their multiplication by scalars;
for example,
f + (g + h) = (f + g) + h,
at + ag = a(f + g),
of + M = (a + b)f.
Matrix methods, especially matrix multiplication, are often
very useful in solving problems. Thus we have applied Equation
513 in Section 27e, after deriving the equation in order to use it.
Another example of the use of this equation is provided by
Problem 512.
In quantummechanical discussions the matrix f corresponding
to the dynamical quantity /(g t , p») is sometimes defined with the
use of the wave functions ^ n , which include the time (Eq. 511),
and sometimes with the wave functions ^ w , with the time factor
1 See, for example, M. B6cher, "Introduction to Higher Algebra," The
Macmillan Company, New York, 1924.
420 GENERAL THEORY OF QUANTUM MECHANICS [XV51a
omitted, in which case the matrix elements are given by the
integrals
fmn = /«/op.Mr. (514)
The matrix elements f mn in the two cases differ only by the time
2x\(WmWn)t
factor e h , and as there is no danger of confusion the
same symbol can be used for the matrix containing the time
as for that not containing the time.
Problem 512. The elements z mn of the matrix x for the harmonic
oscillator are given by Equation 1125. Using the rule for matrix multipli
cation, set up the matrices x 2 ( = xx), x 3 , and x 4 , and compare the values of
the diagonal elements with those found in Section 23a.
The noncommutative nature of the multiplication of matrices
is of great importance in matrix mechanics. The difference of
the product of the matrix q ?  representing the coordinate qj
and the matrix p, representing the canonically conjugate momen
turn pj and the reverse product is not zero, but «— .1, where 1
Jttz
is the unit matrix, discussed in the following section; that is,
these matrices do not commute. On the other hand, q, and
p* (with k t* j)j etc., do commute, the complete commutation
rules for coordinates and momenta being
h <
P/q,  q/P, = 2^1,
p?q*  q*p/ = o, k ?± j,) (5i5)
q>q*  q*q/ = o,
P/P* — p*p? = o.
These commutation rules together with the rules for converting
the Hamiltonian equations of motion into matrix form constitute
matrix mechanics, which is a way of stating the laws of quantum
mechanics which is entirely different from that which we have
used in this book, although completely equivalent. The latter
rules require a discussion of differentiation with respect to a
matrix, into which we shall not enter. 1
Problem 513. Verify the commutation rules 515 by evaluating the
matrix elements (p,g >),»», etc.
1 For a discussion of matrix mechanics see, for example, Ruark and Urey,
"Atoms, Molecules and Quanta," Chap. XVII.
XV61b]
MATRIX MECHANICS
421
51b. Diagonal Matrices and Their Physical Interpretation. — A
diagonal matrix is a matrix whose elements f mn are all zero except
those with m = n; for example,
//oo
/h
/«
. . .
•
/ks ...
The unit matrix, 1, is a special kind of diagonal matrix, all the
diagonal elements being equal to unity:
1 =
A constant matrix, a, is equal to a constant times the unit
matrix:
1
1
1
1
a = al =
a
a
a
. . .
a . . .
Application of the rule for matrix multiplication shows that the
square (or any power) of a diagonal matrix is also a diagonal
matrix, its diagonal elements being the squares (or other powers)
of the corresponding elements of the original matrix.
In Section 10c, in discussing the physical interpretation of the
wave equation, we saw that our fundamental postulate regarding
physical interpretation requires a dynamical quantity / to have
a definite value for a system in the state represented by the wave
function V n only when f r nn is equal to (/» n ) r , for all values of r.
We can now express this in terms of matrices: If the dynamical
quantity f is represented by a diagonal matrix f then this dynamical
quantity has the definite value f nn for the state corresponding to the
wave function V n of the set ¥ , ^i, • • • .
For illustration, let us discuss some of the wave functions which
we have met in previous chapters. The solutions
422 GENERAL THEORY OF QUANTUM MECHANICS [XV51b
/ 2xiWot \
*o\ = h* h ),*i, ' ' '
of the wave equation for any system correspond to a diagonal
energy matrix
Wo
0 JFi— 0—
H = W 2
W
so that, as mentioned in Section 10c, the system in a physical
condition represented by one of these wave functions has a
fixed value of the total energy.
In the case of a system with one degree of freedom no other
dynamical quantity (except functions of H only, such as H 2 )
is represented by a diagonal matrix; with more degrees of freedom
there are other diagonal matrices. For example, the surface
harmonic wave functions Qi m (&)® m (<p) for the hydrogen atom
and other twoparticle systems separated in polar coordinates
(Sees. 19, 21) make the matrices for the square of the total
angular momentum and the component of angular momentum
along the z axis diagonal, these dynamical quantities thus having
definite values for these wave functions. The properties of
angular momentum matrices are discussed in Section 52.
The dynamical quantities corresponding to diagonal matrices
relative to the stationarystate wave functions SP'o, ^i, • • • are
sometimes called constants of the motion of the system. The
corresponding constants of the motion of a system in classical
mechanics are the constants of integration of the classical equa
tion of motion.
Let us now consider a system whose Schrodinger time functions
corresponding to the stationary states of the system are ^ ,
^i, • • • , ^n, • • • . Suppose that we carry out an experiment
(the measurement of the values of some dynamical quantities)
such as to determine the wave function uniquely. Such an
experiment is called a maximal measurement. A maximal
measurement for a system with one degree of freedom, such as
the onedimensional harmonic oscillator, might consist in the
accurate measurement of the energy; the result of the measure
ment would be one of the characteristic energy values W n \ and
the corresponding wave function ^ n would then represent the
XV51b] MATRIX MECHANICS 423
system so long as it remain undisturbed and could be used for
predicting average values for later measurements (Sees. 10, 12;.
A maximal measurement for a system of three degrees of freedom,
such as the threedimensional isotropic harmonic oscillator or
the hydrogen atom with fixed nucleus and without spin, might
consist in the accurate determination of the energy, the square
of the total angular momentum, and the component of the
angular momentum along the z axis. The wave function cor
responding to such a maximal measurement would be one of
those obtained by separating the wave equation in polar coordi
nates, as was done in Chapter V.
It is found that the accurate measurement of the values of N
independent 1 dynamical quantities constitutes a maximal
measurement for a system with N degrees of freedom. In
classical mechanics a maximal measurement involves the accurate
determination of the values of 2N dynamical quantities, such as
the N coordinates and the N momenta, or for a onedimensional
system the energy and the coordinate, etc. A discussion of the
significance of this fact will be given in connection with the
uncertainty principle in Section 53.
Now let us consider a complete set of orthogonal normalized
wave functions xo, Xi> * * * > Xn', • • • , each function Xn' being
a solution of the Schrodinger time equation for the system under
discussion. These wave functions are linear combinations of
the stationarystate wave functions ^ n , being obtained from
them by the linear transformation
Av = ^a n > n * n , (516)
n
in which the coefficients a n > n are constants restricted only in
that they are to make the x's mutually orthogonal and normal
ized. A set of wave functions Xn' is said to form a representation
of the system. Corresponding to each representation matrices 2
f, g', etc. can be constructed for the dynamical quantities
/, g, etc., the elements being calculated by equations such as
fm>n> = JXm'/op.Xn'dT (51~7)
1 The meaning of independent will be discussed later in this section.
2 We use primed symbols to indicate that the matrices correspond to the
representation xn f 
424 GENERAL THEORY OF QUANTUM MECHANICS [XV61b
or obtained from the matrices f, g, etc. (corresponding to the
stationarystate representation ^ n ), by the use of the coefficients
a n > n of Equation 516.
So far we have discussed the measurement of constants of the
motion of the system only; that is, of quantities which are repre
sented by diagonal matrices relative to the Schrodinger wave
functions >Po, ^\, ^2, • • • > and which are hence independent of
the time. But in general we might make a maximal measure
ment consisting in the accurate measurement of N dynamical
quantities /, g*, etc., whose matrices f, g, etc., relative to ^ ,
#1, • • • , are not all diagonal matrices. In the case of such a
maximal measurement we must specify the time t = t f at which
the measurement is made. An accurate measurement of the
quantities /, g, etc. at the time t = t' requires that at the time
t = t' the matrices f, g', etc. be diagonal matrices. In order to
find the wave function representing the system at times subse
quent to t = t' (so .long as the system remain undisturbed),
we must find the representation Xn* which makes these matrices
diagonal at the time t = t'. The accurate values of /, g f etc.
obtained by measurement will be identical with the numbers
fn'n'y g n 'n', etc., occurring as a certain diagonal element of the
diagonal matrices f , g', etc., and the wave function representing
the system will be the corresponding Xn'.
It is interesting to notice that the condition that the dynamical
quantity / be represented by a diagonal matrix f ' in the x repre
sentation can be expressed as a differential equation. In order
for V to be a diagonal matrix, /4v must equal for w! not equal
to n' and a constant value, /„'«', say, for m! = n'. This means
that on expanding / op .Xn' in terms of the complete set of functions
X only the one term fn'n'Xw will occur; that is, that
/op.Xn' = /n'«'Xn', (518)
in which f n ' n * is a number, the w'th diagonal element of the
diagonal matrix f. For example, the stationarystate wave
functions V n im for a hydrogen atom as given in Chapter V
satisfy three differential equations,
HopVnlm = Wr&nlm,
1(1 + l)h*
4ir 2
XV62] THE PROPERTIES OF ANGULAR MOMENTUM 425
and
TUf yT, _ m ^ l ^rr
ft* z op. *nZm — "q~ n ^ m >
corresponding to the three dynamical functions whose matrices
are diagonal in this representation ; namely, the energy, the square
of the total angular momentum, and the z component of the
angular momentum. For a discussion of this question from a
different viewpoint see the next section.
62. THE PROPERTIES OF ANGULAR MOMENTUM
As pointed out in the previous section, systems whose wave
equations separate in spherical polar coordinates (such as the
hydrogen atom) possess wave functions corresponding not only
to definite values of the energy but also to definite values of the
total angular momentum and the component of angular momen
tum along a given axis (say, the z axis). In order to prove this
for one particle 1 let us construct the operators corresponding
to M x , My, and M z , the components of angular momentum along
the x, y, and z axes. Since classically
Mx = VPz  ZVy, (52~1)
with similar expressions for M v and M z , the methods of Section
10c for constructing the operator corresponding to any physical
quantity yield the expressions
A,
= 2xA
M x <
My
M t0
2wi\ dx dz/[
( ± ±\
\ X dy y dx)'
2<7tt
In order to calculate the average values of these quantities it is
convenient to express them in terms of polar coordinates. By the
standard methods (see Sec. 16) we obtain
k**(±iAcot*£) (523)
■L'J X op. IE I'lVi. y op. — O *'
1 The total angular momentum and its z component also have definite
values for a system of n particles in fieldfree space; see, for example, Born
and Jordan, "Elementare Quantenmechanik, ,, Chap. IV, Julius Springer,
Berlin, 1930.
426 GENERAL THEORY OF QUANTUM MECHANICS [XV52
and
M ~  ss £ (52 " 4)
We have postulated that the wave equation is separable in
polar coordinates; if we also restrict the potential energy to be a
function of r alone, the dependence of the wave functions on the
angles will be given by
hm n (&, v, r) = e^Mri^iW, (525)
where 0jm(#)<£m(^>) are the surfaceharmonic wave functions
obtained in Section 18. Using these and the expressions in
Equations 523 and 524, we can evaluate the integrals of the
type
M x (l'm'; Im) = fy?'m>nM zv di mn dT. (526)
In order to prove that the square of the total angular momentum
M 2 has a definite value for a given stationary state described by
4>i m n, it is necessary for us to show that the average value of any
power of M 2 is identical with the same power of the average value
(Sec. 10c). By using the properties of matrices given in the
previous section we can considerably simplify this proof. As
stated there, we need only show that Af 2 D . is represented by a
diagonal matrix. Furthermore we can obtain the matrix
for M 2 from the matrices for M x , M VJ and M & by using the relation
defining M 2 in classical mechanics,
M 2 = Ml + M 2 + M 2 } (527)
and applying the rules for matrix multiplication and addition.
If we carry out this procedure, we first find on evaluation of
the proper integrals that
MJ}'m'\ Im) = ^[W + 1)  m(m + l)}^ m ,, m+1 +
{1(1 + 1)  m(m  l))»8 m >, m ~i]8i> th (528)
ih
My(Vm') Im) = ^[{1(1 + 1)  m(m + l)}^, m+1 
{1(1 + 1)  m(m  l)}^,^]^,*, (529)
M,(l'm'; Im) = ~m5 r ,j3 m ', m , (5219)
in which 5 m ', m+ i = 1 for m' = m + 1 and otherwise, etc.
XV52] THE PROPERTIES OF ANGULAR MOMENTUM 427
The next step is to obtain the elements of the matrices M*,
M 2 , and M 2 from these by using matrix multiplication, and then
the elements of M 2 by using matrix addition. The final result
is that M 2 is a diagonal matrix with diagonal elements . 2
It is therefore true that M 2 has a definite value in the state
ypimn] in other words, it is a constant of the motion with the
value «<+££
47T 2
The proof that M z is also a constant of the motion is contained
in Equation 5210, which shows that M* is a diagonal matrix
with diagonal elements mh/2w so that its value is mh/2w for the
state with quantum number m.
Problem 621. Carry out the transformation of Equations 522 into
polar coordinates.
Problem 522. Derive Equations 528, 529, and 5210.
Problem 523. Obtain the matrices for Ml, M\, M\ by matrix multi
plication and from them obtain the matrix for M 2 .
There is a close connection between the coordinate system in
which a given wave equation is separated and the dynamical
quantities which are the constants of the motion for the resulting
wave functions. Thus for a single particle in a spherically
symmetric field the factor S(d, <p) of the wave function which
depends only on the angles satisfies the equation (see Sec. 18a)
4^ //sin fl^f) + J— d ^ 2 = 1(1 + 1)5. (5211)
sin & d&\ ddj sin 2 & dip 2 v ■ / v
It can be shown that the operator for M 2 in polar coordinates
has the form
Ml. = ^j45 A( sin »Si) + ^1i Til' < 52 ~ 12 )
p 47r 2 (sin # d&\ d$/ sin 2 # dip 2 )
so that Equation 5211 may be written
ttu = 1(1 + l)^+ni m , (5213)
since \l/ = S(&, <p)R(r) and M 2 V . does not affect R(r).
Furthermore the equation for $,»(<?), the <p part of ^, is (Sec.
18a)
g = _ m ** f (5214)
428 GENERAL THEORY OF QUANTUM MECHANICS [XV58
whereas from Equation 524 we find that
M  = SS' (52_15)
so that Equation 5213 may be written in the form
M? ov .+ nlm = m*£&ni». (5216)
The formal similarity of Equations 5213 and 5216 with the
wave equation
is quite evident. All three equations consist of an operator
acting upon the wave function equated with the wave function
multiplied by the quantized value of the physical quantity repre
sented by the operator. Furthermore, the operators jf/ p.,
M%^, and AfJ op . will be found to commute with each other;
that is,
H ov .{Mlx) = MlXH ov . x ),
etc., where x is any function of #, <p } and r.
It is beyond the scope of this book to discuss this question more
thoroughly, but the considerations which we have given above for
this special case can be generalized to other systems and other
sets of coordinates. Whenever the wave equation can be
separated it will be found that the separated parts can be thrown
into the form discussed above, involving the operators of several
physical quantities. These physical quantities will be constants
of the motion for the resulting wave functions, and their operators
will commute with each other.
63. THE UNCERTAINTY PRINCIPLE
The Heisenberg uncertainty principle 1 may be stated in the
following way:
The values of two dynamical quantities f and g of a system can be
accurately measured at the same time only if their commutator is
zero; otherwise these measurements can be made only with an
uncertainty AfAg whose magnitude is dependent on the value of the
commutator. In particular, for a canonically conjugate coordinate
1 W. Hbisknbbrg, Z. f. Phys. 43, 172 (1927).
XV63] THE UNCERTAINTY PRINCIPLE 429
q and momentum p the uncertainty AgAp is of the order of rruignitude
of Planck's constant h, as is AWAJ for the energy and time.
To prove the first part of this principle, we investigate the
conditions under which two dynamical quantities / and g can be
simultaneously represented by diagonal matrices. Let these
matrices be f and g', Xn' being the corresponding representation.
The product f'g' of these two diagonal matrices is found on
evaluation to be itself a diagonal matrix, its n'th element being
the product of the n'th diagonal elements f n > and g n > of the
diagonal matrices f and g'. Similarly g'f is a diagonal matrix,
its diagonal elements being identical with those of f'g'. Hence
the commutator of f and g' vanishes: f'g' — g'f = 0. The
value of the right side of this equation remains zero for any
transformation of the set of wave functions, and consequently
the commutator of f and g vanishes for any set of wave functions;
it is invariant to all linear orthogonal transformations. We
accordingly state that, in order for two dynamical quantities f and g
of a system to be accurately measurable at the same time, their com
mutator must vanish; that is } the equation
fg  gf = (531)
nust hold.
A proof of the second part of the uncertainty principle is
lifficult; indeed, the statement itself is vague (the exact meaning
of A/, etc., not being given). We shall content ourselves with
the discussion of a simple case which lends itself to exact treat
ment, namely, the translational motion in one dimension of a
free particle.
The wave functions for a free particle with coordinate x are
2iriV2mW(xxo) 2riWt
Ne h e h (Sec. 13), the positive sign in the first
exponential corresponding to motion in the x direction and the
negative sign in the — x direction. On replacing W by p/2m
2iripx(x — Xg) 2«tpj<
this expression becomes Ne h e 2mh , in which positive
and negative values of the momentum p z refer to motion in the x
direction and the —x direction, respectively. A single wave
function of this type corresponds to the physical condition in
which the momentum and the energy are exactly known, that
is, to a stationary state of the system. We have then no knowl
edge of the position of the particle, the uncertainty Ax in the
430 GENERAL THEORY OF QUANTUM MECHANICS [XV53
coordinate x being infinite, as is seen from the probability dis
tribution function ty*ty, which is constant for all values of x
between — c© and + oo . When Ap x is zero Ax is infinite.
Now let us suppose that at the time I = we measure the
momentum p x and the coordinate x simultaneously, obtaining
the values p and x , with the uncertainties Ap x and Ax, respec
tively. Our problem is to set up a wave function x which
represents this physical condition of the system One way
of doing this is the following. The wave function
/°° (Px— Po) a 2vipx(x~xo) 2irivlt
j 2(Ap,)> e h e 2mk ( ip x (532)
(PrP0)»
corresponds to a Gaussianerrorcurve distribution e (A/>r) '
of the values of the momentum p x about the average value p ,
with the uncertainty 1 Ap x . (The factor Yi in the exponent in
Equation 532 results from the fact that the coefficients of the
wave functions are to be squared to obtain probability values.)
A is a normalization constant. On evaluating the integral we
find for x at the time t = the expression
_ 27r2(Ap r)H*so) g , 2ytp (3gjo)
x(0) = Be h> + h 9 (53_ 3)
which corresponds to the probability distribution function
for x
X*(0)x(0) = BH W (534)
with
**  sfe (53  5)
This is also a Gaussian error function, with its maximum at
x = x and with uncertainty Ax given by Equation 535. It is
seen that the wave function x corresponds to the value h/2ir
for the product of the uncertainties Ax and Ap x at the time t = 0,
this value being of the order of magnitude h, as stated at the
beginning of the section.
Problem 531. Evaluate the normalization constants A and B 2 by
carrying out the integration over p x and then over x.
1 The quantity Ap x is the reciprocal of the socalled precision index of
the Gaussian error curve and is larger than the probable error by the factor
2.10; see R. T. Birge, Phys. Rev. 40, 207 (1932).
XV53] THE UNCERTAINTY PRINCIPLE 431
Problem 632. Carry through the above treatment, retaining the time
factors. Show that the center of the wave packet moves with velocity
po/m, and that the wave packet becomes more diffuse with the passage of
time.
A gener&l discussion* by the use of the methods of transforma
tion theory (Sec. 54), which we shall not reproduce, leads to the
conclusion that the product of the uncertainties AfAg accompany
ing the simultaneous measurement of two dynamical quantities
/ and g is at least of the order of magnitude of the absolute value
of the corresponding diagonal element in their commutator
fg — gf» (The commutator of x and p z is ^— .1 (Eq. 515), the
absolute value of the diagonal elements being A/27T, in agreement
with the foregoing discussion.) This leads to the conclusion that
the energy W and time t are related regarding accuracy of measure
ment in the same way as a coordinate and the conjugate momen
tum, the product of the uncertainties AW and At being of the
order of magnitude of h (or h/2ir) . In order to measure the energy
of a system with accuracy AW, the measurement must be
extended over a period of time of order of magnitude h/AW.
Problem 533. Show that the commutator Wt  tW has the value
h h A
— — — .1 by evaluating matrix elements, recalling that W ov — and
^Trt ' 2iri dt
top. == *.
It is natural for us to inquire into the significance of the
uncertainty principle by analyzing an experiment designed to
measure x and p x . Many "thought experiments" have been
discussed in the effort to find a contradiction or to clarify the
theory; in every case these have led to results similar to the
following. Suppose that we send a beam of light of frequency
v along the axis AO of Figure 531, and observe along the
direction OB to see whether or not the particle, restricted to
motion along the x axis, is at the point or not. If a light
quantum is scattered into our microscope at B, we know that
the particle is in the neighborhood of 0, and by analyzing the
scattered light by a spectroscope to determine its frequency v f
we can calculate the momentum of the particle by use of the
equations of the Compton effect. But for light of finite fre
quency the resolving power of the microscope is limited, and oui
432 GENERAL THEORY OF QUANTUM MECHANICS [XV54
measurement of x will show a corresponding uncertainty Ax,
which decreases as the frequency increases. Similarly the
measurement of the momentum by the Compton effect will show
an uncertainty Ap x , increasing as the frequency increases.
The detailed analysis of the experiment shows that under the
most favorable conditions imaginable the product AxAp* is of
the order of magnitude of h. l
/
i /
!/
i
i
hV
Source of light
Fia. 531. — Diagram of experiment for measuring x and p x of particle.
64. TRANSFORMATION THEORY
In discussing the behavior of a system the following question
might arise. If at the time t = t' the dynamical property / is
1 For the further discussion of the uncertainty principle see W. Heisenberg,
"The Physical Principles of the Quantum Theory," University of Chicago
Press, Chicago, 1930; N. Bohr, Nature 121, 580 (1928); C. G. Darwin, Proc.
Roy. Soc. A117, 258 (1927); A. E Ruark, Phys. Rev. 31, 311, 709 (1928);
E. H. Kennard, Phys. Rev. 31, 344 (1928); H. P. Robertson, Phys. Rev. 34,
163 (1929); 35, 667.(1930); 46, 794 (1934); and also Ruark and Urey, "Atoms,
Molecules and Quanta," Chap. XVIII; and other references listed at the
end of the chapter.
XVM] TRANSFORMATION THEORY 433
found on measurement to have the value /', what is the prob
ability that the immediately subsequent measurement of the
dynamical property g will yield the value g"! We know one
way to answer this question, namely, to find the wave function 1
X (one of the representation which makes V a diagonal matrix)
corresponding to the value /' of /, to use it to calculate the
average value of all powers of g, and from these to construct a
probability distribution function for g. This is not a very
simple or direct procedure, however; it is of interest that an
alternative method has been found by means of which these
probability distribution functions can be calculated directly.
This method, called the transformation theory* is a general quan
tum mechanics within which wave mechanics is included, the
Schrodinger wave equation being one of a large number of
equations of the theory and the Schrodinger wave functions a
particular type of transformation functions. We shall not
enter into an extensive discussion of transformation theory but
shall give only a brief description of it.
Let us represent by ((/'I/') a probability amplitude function or
transformation function such that (g'\f')*(g'\f) is the probability
under discussion, (g'\f)* being the complex conjugate of (g'\f).
[In case that g' can be any one of a continuum of values,
(g'\f')*(g'\f) is interpreted as a probability distribution function,
the probability that g have a value between g r and g f + dg'
being (g'\f')*(9'\f'W.]
The Schrodinger stationarystate wave functions are proba
bility amplitude functions between the energy and the coordi
nates of the system. For a system with one degree of freedom,
such as a harmonic oscillator, the wave functions \p n are the
transformation functions (x'\W f ) between the coordinate x and
the characteristic energy values, and for the hydrogen atom
the wave functions \l/ n im(r, #, <p), discussed in Chapter V, are the
transformation functions (r^VVto) between the coordinates
r, #, and <p of the electron relative to the nucleus and the charac
teristic energy values W n , the square of angular momentum values
1 In case that the measurement of / is not a maximal measurement many
wave functions might have to be considered.
* The transformation theory was developed mainly by P. A. M. Dirac,
Proc. Roy. Soc. A113. 621 (1927). and P. Jordan, Z.f. Phys. 40, 809 (1927)}
44. 1 (1927).
434 GENERAL THEORY OF QUANTUM MECHANICS [XV54
— j— 2 — ) and the angular momentum component values mh/2Tr 9
represented by the symbols n, l, and m, respectively.
Two important properties of transformation functions are the
following:
The transformation function between / and g is equal to that
between g and /:
(fV) = (g'\n* (54D
The transformation function between / and h is related to that
between / and g and that between g and h by the equation
(f'W) = f(f'\9')*(9'\h'W. (542)
In this equation the integration includes all possible values g'
which can be obtained by measurement of g; in case that g f
represents a set of discrete values, the sum over these is to be
taken.
We have often Written the Schrodinger wave equation in the
form
In the nomenclature of transformation theory this is
W f representing a characteristic value W n of the energy and
(<7;l^') the corresponding transformation function to the coordi
nates g,. In transformation theory it is postulated that a similar
equation
*.,„(/¥) = qV'W) (543)
is satisfied by every transformation function (/%')• In this
equation g ov . is the operator in the / scheme representing
the dynamical quantity g. We shall not discuss the methods
by means of which the / scheme of operators is found but shall
restrict our attention to the q scheme, in which the operators
are obtained by the familiar method of replacing pk by *—.  —
Zti ogic
The transformation functions are normalized and mutually
orthogonal, satisfying the equation
Kf'lffWlf'W = h fr . (544)
XV64] .. TRANSFORMATION THEORY 435
It is interesting to note that this equation signifies that, if the
dynamical quantity / has been found on measurement to have
the value /', immediate repetition of the measurement will give
the same value/' with probability unity, inasmuch as the integral
of Equation 544 is the transformation function (/'I/") (see
Eq. 542) and Equation 544 requires it to vanish except when
/" is equal to/', in which case it has the value 1.
From the above equations we can find any transformation
function (/'#'), using the q system of operators only, in the
following way: we find the transformation functions (q'\f) and
WW) by solving the corresponding differential equations
543, and then obtain (/%') by integrating over the coordi
nates (Eq. 544). As an example, let us obtain the transforma
tion function 0^1^') between the energy W and the linear
momentum p x of a onedimensional system. The function
Or'  IT') is the Schrodinger wave function, obtained by solving
the wave equation
H^(x'\W) = W'(z'\W)
as described in the preceding chapters of the book. The trans
formation function (x'\p' x ) between a Cartesian coordinate and
its canonically conjugate momentum is the solution of the
equation
P^(*'lPi) = v' x {Av' x )
or
and hence is the function
2tjx' P ' x
(x'\ V ' x ) = Ce » , (545)
C being a normalizing factor. The transformation function
(p' x \W), the momentum probability amplitude function for a
stationary state of the system, is accordingly given by the
equation
2tjx'pZ
(P',\W) = JCe h (x'\W'W (546)
or
2idx'p' x
(P' 9 \W») = fCe * M*'W (547)
436 GENERAL THEORY OF QUANTUM MECHANICS [XV64
On application of this equation it is found that the momentum
wave functions for the harmonic oscillator have the same form
(Hermite orthogonal functions) as the coordinate wave functions
(Prob. 541), whereas those for the hydrogen atom are quite
different. 1
Problem 541. Evaluate the momentum wave functions for the harmonic
oscillator. Show that the average value of p r x for the nth state given by
the equation
/:
t (p.\w»)*(p' M \w,) P ';dp'.
is the same as given by the equation
I>fe)'£
Problem 642. Evaluate the momentum wave function for the normal
hydrogen atom,
(p x P v Vz\^m) = J J JCe h (x'y'z'\nlm)dx'dy'dz'.
It is convenient to change to polar coordinates in momentum space as well
as in coordinate space.
The further developments of quantum mechanics, including the
discussion of maximal measurements consisting not of the
accurate determination of the values of a minimum number of
independent dynamical functions but of the approximate meas
urement of a larger number, the use of the theory of groups, the
formulation of a relativistically invariant theory, the quantiza
tion of the electromagnetic field, etc., are beyond the scope of
this book.
General References on Quantum Mechanics
Matrix mechanics:
M. Born and P. Jordan: "Elementare Quantenmechanik," Julius
Springer, Berlin, 1930.
Transformation theory and general quantum mechanics:
P. A. M. Dirac: ''Quantum Mechanics," Oxford University Press, New
York, 1936.
^he hydrogenatom momentum wave functions are discussed by B.
Podolsky and L. Pauling, Phys. Rev. 34, 109 (1929), and by E. A. Hylleraas,
Z./.P/iy«.74,216(1932).
XV54] TRANSFORMATION THEORY 437
J. v. Neumann: " Mathematische Grundlagen der Quantenmechanik,"
Julius Springer, Berlin, 1932.
Questions of physical interpretation:
W. Heisenbbrg: "The Physical Principles of the Quantum Theory,"
University of Chicago Press, Chicago, 1930.
General references:
A. E. Rtjark and H. C. Urey: "Atoms, Molecules and Quanta,"
McGrawHill Book Company, Inc., New York, 1930.
E. U. Condon and P. M. Morse: "Quantum Mechanics," McGrawHill
Book Company, Inc., New York, 1929.
A. Sommerfeld: "Wave Mechanics," Methuen & Company, Ltd.,
London, 1930.
H. Weyl: "The Theory of Groups and Quantum Mechanics," E. P.
Dutton & Co., Inc., New York, 1931.
J. Frenkel: "Wave Mechanics," Oxford University Press, New York.
1933.
APPENDIX I
VALUES OF PHYSICAL CONSTANTS 1
Velocity of light c = 2.99796 X 10 10 cm sec" 1
Electronic charge e = 4.770 X 10~ 10 abs. e.s.u.
Electronic mass w = 9 . 035 X 10~ M g
Planck's constant h = 6 . 547 X 10~ 27 erg sec
Avogadro's number N = 0.6064 X 10 24 mole" 1
Boltzmann's constant k = 1 . 3709 X 10~ 16 erg deg" 1
2*e 2
Finestructure constant a — = 7.284 X 10""*
he
Radius of Bohr orbit in normal hydro
gen, referred to center of mass a = 0.5282 X 10~ 8 cm
Rydberg constant for hydrogen Zfe = 109677.759 cm 1
Rydberg constant for helium # He = 109722 . 403 cm 1
Rydberg constant for infinite mass R„ = 109737.42 cm 1
h
Bohr unit of angular momentum — = 1 . 0420 X 10"" 27 erg sec
Magnetic moment of 1 Bohr magneton /*o = 0.9175 X 10"" 20 erg gauss"" 1
Relations among Energy Quantities
1 erg  0.6285 X 10 12 v.e. = 0.5095 X 10 16 cm" 1 = 1.440 X 10 18 cal/mole
1.591 X 10" 12 erg = 1 v.e. = 8106 cm" 1 = 23055 cal/mole
1.963 X 10" 18 erg = 1.234 X 10~ 4 v.e. = 1 cm"" 1 = 2.844 cal/mole
0.6901 X 10 18 erg = 4.338 X 10" 6 v.e. = 0.3516 cm" 1 = 1 cal/mole
1 These values are taken from the compilation of R. T. Birge, Rev. Mod.
Phys. 1, 1 (1929), *as recommended by Birge, Phys. Rev. 40, 228 (1932).
For probable errors see these references.
439
APPENDIX II
PROOF THAT THE ORBIT OF A PARTICLE MOVING
IN A CENTRAL FIELD LIES IN A PLANE
The force acting on the particle at any instant is in the direc
tion of the attracting center (see F, Fig. 1). Let the arrow
marked v in the figure represent the direction of the motion at
any instant. Set up a system of Cartesian axes xy z with origin
at the point P and oriented so that the z axis points along v and
the y axis points perpendicular to the plane of F and v, being
directed up from the plane of the paper in the figure.
z
y
Fio. IIl.
Then the equation of motion (in Newton's form) in y is
since there is no component of the force F in the y direction.
Therefore the acceleration in the y direction is zero and the
velocity in the y direction, being initially zero, will remain zero,
30 that the particle will have no tendency to move out of the plane
ietermined by F and v
440
APPENDIX III
PROOF OF ORTHOGONALITY OF WAVE FUNCTIONS
CORRESPONDING TO DIFFERENT ENERGY LEVELS
We shall prove that, if W n ^ Wk, the solution ^„ of the
wave equation
N
2^» + TprV.  V)*. = (l)
and the solution \p£ of the equation
N
t1
satisfy the relation
mtndr = 0; (3)
i.e., that ypk is orthogonal to \p n .
Multiply Equation 1 by \f/*, Equation 2 by ^ n . and subtract
the second from the first. Since V is real, the result is that
N
2^***"*"  *•*#*> + 7? (Tf "  *W*» = o. (4)
t1
If we now integrate the terms of this equation over configuration
space, we obtain
N
~(W %  Wu) fitter = 2^J ( ** V ' V,i " **&*)*• (5)
t1
If we introduce the expression for v» ? in terms of Cartesian coordi
nates into the integral on the right, it becomes
Jl
441
442 APPENDIX III
in which we have written qi, g 2 ? • • • , Qzn in place of x i} y Xi Zi t
Xi,    > «at. We next make use of the identity
'd qi
from which we see that
because of the boundary conditions on ^.
Since every term of the sum can be treated similarly, the
expression 6 is equal to zero and therefore
^V»  Wkij+t+ndr = 0,
from which Equation 3 follows, since W n — Wk ^ U.
If Tf n s Jf*, so that ^fc and ^„ are two linearly independent
wave functions belonging to the same energy level, \p k and \p n are
not necessarily orthogonal, but it is always possible to construct
two wave functions i/^ and \f/' n > belonging to this level which are
mutually orthogonal. This can be done in an infinite number of
ways by forming the combinations
Vk> = c&k + Mn and <A«' = a'+k + 0tyn,
with coefficients a, p, a', ft satisfying the relation
WWn'dr = a*a'S+t+ k dT + a*ft m+ndr + a'pfri+dr +
P*P'm+jT = 0. (8)
APPENDIX IV
ORTHOGONAL CURVILINEAR COORDINATE SYSTEMS
In Section 16 the general formulas for the Laplace operator v 2
and for the volume element dr were given in terms of the quanti
ties q u , q vy and q w defined by Equation 164. In this appendix
there are given the equations of transformation (in terms of
Cartesian coordinates) and the expressions for the q'a for the 11
sets of orthogonal coordinate systems listed by Eisenhart 1 as the
only such systems in which the threedimensional Schrodinger
wave equation can be separable. In addition the explicit
expressions 2 for v 2 and dr are given for a few of the more impor
tant systems. These quantities may be obtained for the other
systems by the use of Equations 163 and 165.
Cylindrical Polar Coordinates
X = p COS <p,
y = p sin <p,
z = z.
q P = 1, q a = 1, q<p = p.
dr = pdpdzdip.
v p d P \%J "■" p 2 d<p 2 "*■ dz 2
Spherical Polar Coordinates
x = r sin # cos <p,
y = r sin & sin <p, (Fig. 11),
z = r cos #.
q r = 1, ^ = r, <k = r sin #.
dr = r 2 sin &drd&d<p.
v2 = T 2 l( r fr) + ra^ l>( 8in * £)
+
1
r 2 sin 2 # d<p 2
1 L. P. Eisenhart, Phys. Rev. 45, 428 (1934).
2 E. P Adams, "Smithsonian Mathematical Formulae," Washington,
1922. This book contains extensive material on curvilinear coordinates as
well as other very useful formulas.
443
444 APPENDIX IV
Parabolic Coordinates
X = \/^7 COS (p,
y = Vfr sin <p,
«/€ = 2\ — F""' ft = 2\"~V~ ; ft = vw
d T = /4(£ + y)d£dT)d<p.
2 = _4__ _*/ 1*^ , _i_ 1/ l\ , 1 i!
v £ + v a*\ w * + * M%/ fr a^ 2 '
Confocal Elliptic Coordinates (Prolate Spheroids)
x = as/^ 2 — 1\/1 — *7 2 cos v?,
2/ = a\/£ 2 — 1a/1 ~~ *? 2 sin <p,
z = a^.
In terms of the distances r A and r B from the points (0, 0, — a)
and (0, 0, a), respectively, £ and rj are given by the expressions
t _ r A + r B r A  r B
5 ~ 2a ; ^ 2a
«« = a V5^r' * = Wf=?' ft = aV( * 2  Da  r? 2 ).
dr = a 3 (£ 2  r} 2 )d&r)d<p.
}
+
e  v*
(e  D(i  r, 2 ) d*>*J
Spheroidal Coordinates (Oblate Spheroids)
x = a£j cos ip, y = a£»> sin ?,« = a\/(£ 2  1)(1  v 2 ).
ft  a \ ^2 Z i ' 9 = a \j l H v 2 > 9* = aft
Parabolic Cylinder Coordinates
x = %(u  v), y  Vuv, z =2.
. 1 ju + v n 1 \ u + v
? " = 2\T"' qv = ^V  ^' 9 * = L
APPENDIX IV 445
Elliptic Cylinder Coordinates
x = ay/(u 2 — 1)(1 — v 2 ), y = auv, z = z.
\u 2  v 2 \u 2  V 2
Qu = a yl u 2  i ' Qv = a \ i  v * ' q ° =
Ellipsoidal Coordinates
2 _ (a 2 +^)( a 2 + t; )(a2 + ^) 2 _ (b* + u)(b* + v)(b* + w)
X (a 2  6 2 )(a 2  c 2 ) ' y (6 2  c 2 )(b 2  a 2 ) '
., = (c 2 + u)(c 2 + v)(c 2 + w)
(c 2  a 2 )(c 2  b 2 )
*2 =
2 = (u  v)(u  w) 2 (v  ti?)(ty  u)
qu 4(a 2 + u)(b 2 + u)(c 2 + u)' q * 4(a 2 + v)(b 2 + v)(c 2 + v)'
ft =
(w — w)(w — v)
4(a 2 + w)(6 2 + w)(c 2 + w)
Confocal Parabolic Coordinates
x = ^{u + v + w  a  6), y 2 = fr  fl /
t (6  u)(6  v)(b w) ^ , . . .
a — b
2 _ ( M — ^)(^ — w) 2 _ (P — U)(V — to)
** ~ 4(o  u)(6  u)' Qv ~ 4(a  v)(fi  v)'
2 — ( W ~ U )( W ~ V )
qw  4(a  w)(6  tu)'
A Coordinate System Involving Elliptic Functions
x = u dn(t>, k) sn(w, fc'), 2/ = u sn(t;, &) dn(w, A;'),
z = u cn(t>, Aj) cn(w, &'), k 2 + A;' 2 = 1.
ql = 1, ffj = tfi = ^ 2 {fc 2 cn 2 (;;, t) + A;' 2 cn 2 (w, *')}.
For a discussion of the elliptic functions dn, sn, and en see
W. F. Osgood, "Advanced Calculus," Chap. IX, or E. P. Adams,
"Smithsonian Mathematical Formulae" p. 245.
APPENDIX V
THE EVALUATION OF THE MUTUAL ELECTROSTATIC
ENERGY OF TWO SPHERICALLY SYMMETRICAL
DISTRIBUTIONS OF ELECTRICITY WITH
EXPONENTIAL DENSITY FUNCTIONS
In Section 23b there occurs the integral
Ze 2 f Ce^ur^n. .
 I I —7 ClTidT 2,
327r 2 aoJ J P12
in which pi = 2Zri/a and dri = p 2 dpi sin &id&id<pi, with similar
expressions for p 2 and dr 2 , n, #1, <Pi and r 2 , #2, <p 2 being polar
coordinates for the same system of axes. The quantity P12
represents 2Zri 2 /a , in which r i2 is the distance between the
points r x , #1, <pi and r 2 , # 2 , <p 2 .
This integral (aside from the factor Ze 2 /327r 2 a ) represents the
mutual electrostatic energy of two spherically symmetrical
distributions of electricity, with density functions e~ p i and
e~ p *, respectively. It can be evaluated by calculating the
potential due to the first distribution, by integrating over
dri, and then evaluating the energy of the second distribution
in the field of the first.
The potential of a spherical shell of radius pi and total charge
btrp\er p xdpi is, 1 at a point r,
and
4irple p idpi • — for r < pi
Pi
4rp 2 i e~ p idp 1 •  f or r > pi;
r
that is, the potential is constant within the shell and has the
same value outside of the shell as if the entire charge were
located at the origin.
1 See, for example, Jeans, "Electricity and Magnetism," Cambridge
University Press, Cambridge, 1925, Sec. 74.
446
APPENDIX V 447
The potential of the complete distribution is hence
<t>(r) = — I e p ip\dpi + 4tt I e~ p ipidpi,
which is found on evaluation to be
*(r) =^{2e(r + 2)}.
The integral / then has the value
1 = 32^J *<")^«*t,
Ze 2 f "
= 2^1 {2  e'^pj + 2)je"»p 2 rf P8 ,
which gives on integration
APPENDIX VI
NORMALIZATION OF THE ASSOCIATED LEGENDRE
FUNCTIONS
We can obtain the orthogonality property of the functions
P\ m] (z) and P\T^(z) as follows: Multiply the differential equation
199 satisfied by P\ m] (z) by P[? ] (z) and subtract from^his the
differential equation satisfied by P\? ] (z) multiplied by P[ m{ (z).
The result is the relation
= \v<y + l)  l(i + i)}P\rP[ m K
If we integrate this between the limits —1 and 1, we obtain the
result
{l'(V + 1)  1(1 + \))f^P\7*{z)P\*(z)dz
Therefore, if V ^ I,
j^Pf(z)P\^(z)dz = 0. (1)
This result is true for any value of m, so it is also true for the
Legendre functions Pi(z), since Pi(z) = P?(z).
We can now obtain the normalization integral for the Legendre
polynomials. Replacing I by I — 1 in Equation 192 gives the
equation
Pi(z) = j{(2l  i)riViOO  (J  DPitOO).
Using this and the orthogonality property just proved, we obtain
the relation
448
APPENDIX VI 449
J + 1 271 f + 1
i {Pi{z))Hz = fi T  i J_ i Pii(z)zPi(z)dz.
Equation 192 can be written in the form
zPiiz) = 2TT1 {(Z + 1)Pz + l(z) + lPl ~^>
so that, again employing the orthogonality property, we get
This process can be repeated until the relation
x
{iM*)P<k =
(21  l)(2l  3)(2l  5) • • 31 f +1
(21 + l)(2l  1)(21  3) ■ • • 5 3j_i ,r,w '
= 2nnXr {Po(i5)12<fo
is obtained. PoOO is by definition (Eq. 191) the coefficient of
t° in the expansion of (1 — 2tz + t 2 )~~M in powers of t. It is
therefore equal to unity, so that
£>K*)^ = ^£> =
21 + 1
(2)
To obtain the normalization integral for the associated
Legendre functions we proceed as follows. 1 By differentiating
Equation 197 and multiplying by (1 — z 2 )^ we obtain
(1  ^Mfi^al = (1  ^ 2 I^P^)"
Ml J«
M*d  * 2 ) 2 ^ra p '(«) = p l ml+1 (2)  M*(i  « 2 )^l ml («).
I 2
+
Transposing, squaring, and integrating gives
xr {pim,+i(2)} ^=xr[ (i ^{^} 2
1 Whittaker and Watson, "Modern Analysis," Sec. 1551.
450 APPENDIX VI
f + 1 mW
+ Ji 1 2
where integration by parts 1 has been employed to obtain the
first two terms of the last line.
If we now use the differential equation 199 for P\ ml (z) to
reduce the first term of the last line, we obtain, after combining
terms, the result
f^iPP+KWdz = (I  M)(Z + \m\ + l)f^{P\Kz)} 2 dz.
We can continue this process and thus obtain
f^{P\ m Kz)} 2 dz = (1  \m\ + l)(l  \m\ + 2) • • • I
(I + \m\)(l + m  1) • • • (Z + l)f +*&&)}*&,
so that
x
i {n {)] 21 + 1(1 \m\)f
where we have used the result of Equation 2.
/r dP i r*
udv — uv — I vdu, we set u = (1 — z 2 )— — ,
dP\ mS
dv — in order to reduce the first term, and u = z,
dz
jpm
dv  2Pl m — l —dz « d(P' z m, p
dz
to reduce the second term. The term in uv vanishes, In the first case because
(1 — z 2 ) is zero at the limits, and in the second case because P[ m (2) is zero
at the limits, if m ^ 0.
APPENDIX VII
NORMALIZATION OF THE ASSOCIATED LAGUERRE
FUNCTIONS
In order to obtain Equation 2010, we make use of the generat
ing function given in Equation 208, namely
^^(pLr.riv.
1u
u.( P , u)  2i~W ur s ^'( lu)" "'
r = «
Similarly let
p»
y.y, .)  2^ "  (v ( T^.
* = «
Multiplying these together, introducing the factor e~ p p 8+1 f and
integrating, we obtain the equation
00
J[ Vv+><7.(p, u)F £ ( p , t,)dp = ^^§~ e' P >"L' r {p)L t { P )dp
= (1  w )«+» = ( * + 1)!(1 ~ w ~ " + ™>)
^(g + fc+1)!
where we have expanded (1 — uv)~"~ 2 by the binomial theorem. 1
The integral we are seeking is (r!) 2 times the coefficient of
(uv) r in the expansion, which is
/• 00
x For the value of the integral I p*+ l e~ aP dp see Peirce's " Table of
Integrals."
451
452 APPENDIX VII
We, + i)ii < r + 1 > ! + zl 1
V.) ^ t i;.^ (r _ s)(s + 1}1 f (r _ s _ 1)f(s + 1)f 
(r!) 3 (2r  g + 1)
(r  s) !
In order to obtain the integral of Equation 2010 we must put
r = n + J and s = 21 + 1, yielding the final result
x
'ev<+w + v(p)p<* P = ( yi w +.ffi
APPENDIX VIII
THE GREEK ALPHABET
A, a .
. . Alpha
N, v .
. . Nu
B,0 .
. . Beta
s, f .
. Xi
r, 7 •
. . Gamma
0,0 .
. Omicron
A, 5 .
. . Delta
n, t .
. Pi
E,« .
. . Epsilon
p,p .
. BAo
z,r •
. . Zeta
2,<r .
. Sigma
H, v ■
. . Eta
T,r
. Taw
0, #,
. . Theta
T, v .
. Upsilon
I,i .
. . Iota
% <p 9 4 •
. Phi
K, K .
. . Kappa
X, x • •
. CW
A, X .
. . Lambda
*,* •
. Psi
M,/. .
. . Tkfw
12, co .
. Omega
453
INDEX
Absorption of radiation, 21, 299
Accessible wave functions, 396, 397
Action, 25
Action integrals, 29
Activation energy, 412
Adams, E. P., 201
Adsorption, unactivated, 388
Alkal^atom spectra, £07
Alternating intensities in band spec
tra, 356
Amplitude equation, 56
in three dimensions, 86
Amplitude functions, definitions of
58
Amplitudes of motion, 286
Anderson, C, 209
Angular momentum, of atoms, 237
conservation of, 11
of diatomic molecules. 265
of elefnon spin, 208
of h^pgen atom, 147
properties of, 425
of symmetrical top molecule, 280
Antisymmetric wave function, defi
nition of, 214
Approximate solution of wave equa
tion, methods of, 191
(See also Wave functions.)
Approximation by difference equa
tions? 202
Aromatic carbon compounds, ener
gies of, 379
Associated Laguerre functions,
normalization, 451
polynomials, 131
table of, 135
Associated Legendre functions, 127
table of, 134
Asymptotic solution of wave equa
tion, 68
Atanasoff, J. V., 228
Atomic energy states, semiempirical
treatment, 244
Atomic terms, Hund's rules for, 246
Atomic wave functions, 250
Atoms, with many electrons, 230$".
variation treatments for, 246
Average values, in quantum mechan
ics, 89
of dynamical quantities, 65
of r* for hydrogen atom, 144
Azimuthal quantum number, 120
B
Bacher, R. F., 258
Balmer formula, 27
Balmer series, 43
Bartholomew E., 310
Bartlett, J. H., Jr., 254
Beach, J. Y., 385
Bear, R. S., 415
Beardsley, N. F., 249
Benzene, plane structure of, 411
structure of, 378
Beryllium atom, wave functions for,
249
Bichowsky, R., 208
Birge, R. T., 41, 336, 439
Black, M. M., 254
Black body, 25
B6cher, M., 419
Bohr, N., 26, 36, 112
Bohr frequency rule, 27
Bohr magneton, 47
Bohr postulates, 26
Boltzmann distribution law, 399
Boltzmann statistics, 219
455
456
INDEX
Bonds, chemical, types of, 362
Bond wave functions, 374
Bonhoeffer, K. F., 358
Born, M., 49, 51, 112, 260, 364, 416,
417, 425, 436
BornOppenheimer principle, 260
Bose, S. N., 403
BoseEinstein distribution law, 404
Bose Einstein statistics, 219, 402
Bowen, I. S., 208
Brackett series, 43
Bragg equation, 35
Brester, C. J., 290
Brillouin, L., 198, 412
Brockway, L. O., 379
Brown, F. W., 254
Burrau, , 333, 340
Byerly, W. E., 24
Canonical form of equations of
motion, 16
Canonical set of structures, 375
Carbon atom, tetrahedrai, 364, 377
variation function for, 249
Carbon compounds, aromatic, ener
gies of, 379
Catalysis of orthopara conversion,
358
Centralfield approximation for
atoms, 230, 250
Characteristic energy values, 58
Characteristic functions, definition
of, 58
Characteristic value equation, defi
nition of, 58
Characteristic values, approximate,
180
Chemical bonds, types of, 362
Classical expressions, significance of,
55
Classical mechanics, 2ff.
as an approximation to quantum
mechanics; 198
and high quantum states, 76
Classical statistics, 219
Coefficient, of absorption, definition
of, 300
of induced emission, definition of,
300
of spontaneous emission, defini
tion of, 300
Coefficients in secular equation ,for
molecule, 376
Commutation rules, 420
Completed shells of electrons, 234
Completeness of sets of orthogonal
functions, 154
Complex conjugate wave function,
63, 88
Complex molecules, 366^.
Component of angular momentum,
definition of, 12
Compton, A. H., 35 f
Conditionally periodic systems, defi
nition of, 29
Conditions on wave functions, 58
Condon, E. U., 54, 82, 108, 246, 258,
310, 312, 432
Configuration, electronic, definition
of, 213
Configuration space, definition of,
59
Conjugate wave function, complex,
63
Conservation, of angulaaenomen
tum, 11
of energy in quantum mechanics,
75
Conservative system, definition of , 16
Constant of the motion, definition
of, 12, 422
Continuous sets of energy levels, 58
Coolidge, A. S., 188, 249, 349, 353,
364, 374, 415
Coordinates, curvilinear, 103
cyclic, 108
generalized, 6
ignorable, 108
for molecules, 389
normal, definition of, 287
Correspondence principle, 29
Coulomb integral, 212, 371
Coupled harmonic oscillators, 397
INDEX
457
Courant, R., 91, 120, 157, 192, 202
Cross, P. C, 282
Crystal, diffraction by, 34
Crystals, rotation of molecules in,
290
Curvilinear coordinates, 103
Cyclic coordinates, 108
D
Darrow, K. K, 54, 83, 403, 412
Darwin, C. G., 209, 210, 432
De Broglie, L., 49, 93
De Broglie wave length, 35
Debye, P., 26, 383, 408, 412
Degeneracy, exchange, 230
>^jtal, 367
ipatial, 233
flip 367
Degenerate energy levels, 73, 166
Degenerate states, 31, 100
Degeneration of gases, 405
Del squared, operator, 85
Dennison, D. M., 275, 279, 293, 357
Determinants, certain properties of,
174
Determjuianttype wave functions,
21^232
DiagdHporm for secular equation,
Diagonal matrices, 421
Diagonalsum theorem, 239
Diatomic molecule in old quantum
theory, 32
rotation and vibration of, 263
selection rules and intensities for,
309
Dickinson, B. N., 331
Dieke, 6. I., 282
Dielectric constant, 408
of diatomic dipole gas, 408
and polarizability, 227
Difference equations approximating
wave equation, 202
Differential equation for Legendre
polynomials, 127
standard form for, 109
Diffraction by a crystal in old
quantum theory, 34
Dipole interaction, 384
Dipole moment, electric, definition
of, 303
Dipole radiation, definition of, 23
Dirac, P. A. M., 49, 112, 209, 210,
256, 294, 299, 403, 417, 433, 436
Dirac equations and electron spin,
209
Directed valence, 377
Discrete sets of energy levels, 58
Dissociation energy, of hydrogen
molecule, 349, 352
of hydrogen moleculeion, 336
Distribution law, Boltzmann, 399
Doi, S., 179
Doublets in alkali atom spectra, 207
Duane, W., 35
Dunham, J. L., 198
Dunkel, M., 346
Dunn, C. G., 254
E
Eckart, C, 49, 180, 222, 247, 275,
417
Edlen, B., 225
Ehrenfest, P., 36
Eigenfunction, definition of, 58
Eigenwert, definition of, 58
Einstein, A., 25, 300, 403
Eisenhart, L. P., 105
Eisenschitz, R., 386
Electric dipole moment, definition
of, 303
Electric moment, of molecules, 411
of a system, definition of, 23
Electron, spinning, 207
Electron affinity of hydrogen, 225
Electron densities for atoms> 257
Electron density for lithium, 249
Electron diffraction by a crystal, 34
Electron distribution for hydrogen
moleculeion, 337
Electron distribution function for
lithium, 249
Electronpair bond, 362
458
INDEX
Electronpairing approximation, 374
Electronspin functions for helium,
214
Electronspin quantum number, 208
Electronic configuration, definition
of, 213
Electronic energy function for dia
tomic molecules, 266
Electronic energy of molecules, 259
Electronic states, even and odd, 313
Electronic wave function for mole
cule, 261
Elliptic orbit, equation of, 38
ElSherbini, M. A., 179
Emde, 343
Emission of radiation, 21, 299
Empirical energy integrals for
atoms, 244
Energy, of activation, 412
of classical harmonic oscillator, 5
correction to, firstorder, 159
secondorder, 176
and the Hamiltonian function, 16
of hydrogen moleculeion, 336
kinetic, definition of, 2
of molecules, separation of, 259
potential, definition of, 2
of resonance in molecules, 378
of twoelectron ions, 225
values of, for atoms, 246
Energy level, lower limit for, 189
lowest, upper limit to 181
Energy levels, 58
approximate, 180
for diatomic molecule, 271, 274
for harmonic oscillator, 72
for plane rotator, 177
for symmetrical top molecule, 280
vibrational, of polyatomic mole
cule, 288
Epstein, P. S., 36, 179, 191
Equation, homogeneous, 60
Equations of motion, in Hamil
tonian form, 14
in Lagrangian form, 8
Newton's, 2
Ericson, A., 225
Ethane molecule, free rotation in,
280
Eucken, A., 26
Eulerian angles, 276
Even and odd electronic states, 313
Even and odd states of molecules,
354
Even and odd wave functions for
molecules, 390
Exchange degeneracy, 230
integral, 212, 372
Excited states, of helium atom, 225
of hydrogen molecule, 353
of hydrogen moleculeion, 340
and the variation method, 186
Exclusion principle, 214
Expansion, of l/r»y, 241
in powers of h, 199
in series of orthogonal fun<Maoi6,
151
Eyring, H., 374, 376, 414
Factorization of secular equation
for an at uii, 235
Farkas, A., 358, 414
Fermi, E., 257, 403
FermiDirac distribution law. *ii3
FermiDirac statistics, 219
Field, self consistent, 250#
Fine structure, of hydrogen spec
trum, 207
of rotational bands, alternating
intensities in, 356
Finkelstein, B. N., 331
Fock, V., 252, 255
Force, generalized, 7
Force constant, definition of, 4
Forces between molecules, 3$3
Formaldehyde, rotational fine struc
ture for, 282
Formulas, chemical, meaning of, 380
Fourier series, 153
Fowler, R. H., 412
Franck, J., 310
FranckCondon principle, 309
Frank, N. H., 275
INDEX
459
Free particle, 90
Free rotation in molecules, 280, 290
Frenkel, J., 83, 437
Frequency, of harmonic oscillator, 5
of resonance, 320
Friedrichs, 202
Fues, E., 274
Fundamental frequency, definition
of, 290
G
g factor for electron spin, 208
Geib, K. H., 414
General solution of wave equation,
57
Gerifiral theory of quantum mechan
] ics, 416jf.
Generalized coordinates, 6
GeneHdized forces, 7
Generalized momenta, definition of,
14
Generalized perturbation theory,
191
Generalized velocities, 7
Generating function,' IJfPr'associated
Laauerre polynomials, 131
for aitociate *, Legendre functions,
deMkn of, 77
for^Haerre polynomials, 129
for I^ejjendre polynomials, 126
Gentile, G., 361
Ginsburg, N., 246
Gordon, W., 209
Goudsmit, S., 207, 208, 213, 221,
227, 237, 246, 257, 258, 313
Gropper, L., 405
GroupJ completed, of electrons, 125
deration of, 231
Group theory and molecular vibra
tions, 290
Guillemin, V., 247, 332, 353
Halfquantum numbers, 199
Hamiltonian equations, 16
Hamiltonian form of equations of
motion, 14
Hamiltonian function, definition of,
16
and the energy, 16
and the wave equation, 54
Hamiltonian operator, 54
Harmonic oscillator, average value
of x\ 161
classical, 4
in cylindrical coordinates, 105
energy levels for, 72
in old quantum theory, 30
perturbed, 160
selection rules and intensities for,
306
threedimensional, in Cartesian
coordinates, 100
wave functions, mathematical
properties of, 77
in wave mechanics, 67jf.
Harmonic oscillators, coupled, 315Jf.,
397
Harmonics, surface, 126
Harteck, P., 358, 414
Hartree, D. R., 201, 224, 250, 254,
255
Hartree, W., 255
Hass6, H. R., 185, 228, 385, 387
Heat, of activation, 412
of dissociation, of .hydrogen mole
cule, 349, 352
of hydrogen moleculeion, 336
Heat capacity, of gases, 408
of solids, 26
Heats of sublimation, 388
Heisenberg, W., 48, 112, 209, 210,
226, 318, 416, 417, 428, 432, 437
Heisenberg uncertainty principle, 428
Heitler, W., 340, 361, 364
Helium, solid, equilibrium distance
in, 362
Helium atom, 210
accurate treatments of, 22
excited states of, 225
ionization potential of, 221
normal state of, by perturbation
theory, 162
460
INDEX
Helium atom, polarizability of, 226
resonance in, 321, 324
with screening constant func
tion, 184
spin functions of, 214
stationary states of, 220
Helium moleculeion 358, 367
Hermite orthogonal functions, 80
Hermite polynomials, 77, 81
recursion formula for, 71
Hilbert, D., 91, 120, 157, 192
Hill, E. L., 83
Hiyama, S., 179
Homogeneous equation, definition
of, 60
Homogeneous linear equation, solu
tion of, 169
Hooke's forces in molecules, 282
Hooke's law constant, 4
Hooke's potential energy for dia
tomic molecules, 267
Horowitz, G. E., 331
Houston, W. V., 221
Huckel, E., 346, 365, 379, 381
Hultgren, R., 377
Hund, F., 346, 381, 390
Hund's rules for atomic terms, 246
Hydrogen atom, 112
continuous spectrum of, 125
electron affinity of, 225
energy levels of, 42, 124
momentum wave functions of,
436
normal state of, 139
in old quantum theory, 36Jf.
oldquantumtheory orbits, 43
perturbed, 172
polarizability of, 185, 198, 205
selection rules for, 312
solution of r equation, 121
solution of theta equation, 118
solution of wave equation, 113
spectrum of, 42
Stark effect of, 178, 195
Hydrogen atoms, three, 368, 414
limiting cases for, 372
wave functions for, 368
Van der Waals forces for, 384
Hydrogen chloride, absorption band
of, 33
Hydrogenlike radial wave functions,
discussion of, 142
Hydrogenlike wave functions, 132
discussion of angular part of, 146
Hydrogen molecule, 340Jf.
excited states of, 353
Hydrogen moleculeion, 327j(f.
Hydrogen spectrum, fine structure
of, 207
Hydrogen sulfide, rotational fine
structure for, 282
Hylleraas, E. A., 222, 225, 226, 335,
337, 340, 353, 436
Hypergeometric equation, 278
Identity operation, definition of, 231
Ignorable coordinates, 108
Independent sets of wave functions,
216
Index of refraction and polarizabil
ity, 227
Indicial equd ion, 109
Induced emission, 300 i
Infinite determinant*, soljpon of,
339
Infinity catastrophe, 60
Inglis, D. R., 246
Ingman, A. L., 224
Integrals, energy, for atoms, 239
involved in molecularenergy cal
culations, 370
involving determinanttype wave
functions, 239
Intensities, for diatomic molecule,
309
for harmonic oscillator, 30p
for surfaceharmonic wave func
tions, 306
Interaction, of helium atoms, 361 x
of hydrogen atom and molecule,
373
Interatomic distance in hydrogen
molecule, 349
INDEX
461
Interchange integrals, definition of,
212
Invariance of equations of motion, 7
Inverse permutation, definition of,
231
Ionic 'contribution to bonds, 364
Ionic structures for hydrogen mole
cule, 345
Ionization energy of twoelectron
s "ions, 225
Ionization potential for helium, 221
Ionization' potential for lithium, 247
Ishida, Y., 179
Islands, 376
Ittmann. G. P., 282
JaiHi G., 335, 340
Jahjke, E., 343
James, H. M., 188, 249, 333, 349,
353, 362, 374, 415,
Jeans, J. H., 24, 241
Johnson, M„, / T 5
Johnson, M. h!, Jr.;
Jordlfe P., 49, 112, &09, 416, 417,
433, 436
2,Soa 4
K
£arman, T. von, 26
om, W. H., 383
ku!6 structures, resonance of, 378
ner, G. W., 222
Kemble, E. C., 83, 353
Kennard, E. H., 432
Kerr's area law, 37
KifflLll, G. E., 203, 376, 415
Kine% .^^rgy, definition of, 2
Kirkwood/lKjL, 191, 228, 385, 387
XistiakowsB^ltB . , 282
Klein, 0., 282 W
Kohlrausch, K^jflT. F., 293
Kramers, H. A., 198, 282
Kronig, R. de L., 276, 293, 390
La Coste, L. J. B., 280
Lagrange's equations of motion, 8
Lagrangian function, definition of, 3
Laguerre polynomials, 129
Atype doubling, 390
LandS, A., 208
g factor for electron spin, 208
Langevin, P., 412
Laplace operator, in Cartesian coor
dinates, 85
in curvilinear coordinates, 104
Legendre functions, 125
LennardJones, J. E., 191, 206, 290,
340, 381, 388
Lewis, G. N., 340, 377
Lewy, H., 202
Light (see Radiation)
Linear combinations and resonance,
320
Linear independence of wave func
tions, definition of, 166
Linear momentum, average, of elec
tron in hydrogen atom, 146
Linear variation functions, 186
Lithium atom, electron distribution
function for, 249
wave functions for, 247
London, F., 340, 361, 364, 383, 386,
387, 388, 413
Loney, S. L., 24
Lyman, T., 222
Lyman series, 43
M
MacDonald, J. K. L., 188, 189, 353
McDougall, J., 254
MacMillan, W. D., 24
Magnetic moment, 412
of electron spin, 208
of hydrogen atom, 147
orbital, 47
Magnetic quantum number, 40, 117
Magnetic susceptibility, 412
Magneton, Bohr, 47
Majorana, E., 353, 359
462
INDEX
Manyelectron atoms, 230/.
Margenau, H., 386, 387
Matossi, F., 293
Matrices, 417jf.
Matrix algebra, 417
Matrix mechanics, 416jf.
Maximal measurement, 422
Mayer, J. E., 229
Mayer, M. G., 229
Measurements, prediction of results
of, 66
Mecke, R., 282
Millikan, R. A., 208
Modes of vibration of molecules, 287
Mole refraction, definition of, 227
Molecular energy levels, 259
Molecular orbitals, 346
method of, 381
Molecular wave functions, sym
metry properties of, 388
Molecule, diatomic, selection rules
and intensities of, 309
Molecules, complex, 366jf.
diatomic, rotation and vibration
of, 263
polyatomic, rotation of, 275
vibration of, 282
quantum number A in, 390
Moment of inertia, 269, 275
Momenta, generalized, definition of,
14
Momentum, angular, conservation
of, 11
average linear, of electron in
hydrogen atom, 146
operator, 54
Momentum wave functions, 436
Morse, P. M., 54, 82, 108, 249, 272,
312, 340, 437
function for diatomic molecules,
271
Mott, N. F., 83
Mulliken, R. S., 346, 381
Multiplication of permutations, defi
nition of, 231
Multiplicity of atomic terms, 220
N
Negative states, 392
Nernst, W., 26
Neumann, J. v., 437
Newton's equations, 2
Nielsen, H. H., 280, 282
Niessen, K. F., 327
Nondegenerate energy levels/defini
tion of, 73
Normal coordinates, 282
mode of vibration ©^definition
of, 287
Normalization, of amplitude func
tions, 89
of wave functions, 64
for a continuum, 92
Nuclei r s7pin for hydrogen, out
Nu t'ca# mmetry of electronic irave
i vjtions for molecules, 39t r
Nuclear wave function for molecule,
263
Numerical integration, 201
Old quantum diieory, as an approxi
mation to I quantum mectyjuiics,
198
decline of, 48
Oneelectron bond, 362 1
Operator, for Hamiltonian>Y
for momentum, 54
Operators for dynamical quantiti<
66
Oppenheimer, J. R., 260
Orbit, classical, of threedimen
sional oscillator, 11
Orbital, definition of, 137
Orbital degeneracy, 367
Orbitals, molecular, 381
Orbits, significance of, intouantum
mechanics, 141
Ortho helium, 221
Ortho hydrogen, 35JJW6&
Orthogonal curvilralar coordinate
systems, 441 >,
Orthogonal functions, a convenient
set of, 195
IhDEX
463
Orthogonal functions, expansions in
terms of, 151
Orthogonal transformation, defini
tion of, 288
prthogonality of wave functions, 64,
89, 441
illation of molecules in crystals,
Dr, classical, in polar coor
es, 9
tl harmonic, 4
LarmcSnjfc, K in cylindrical coor
dinates, 105
in old quantum theory, 30
perturbed, 160
tjireedimensional, in Cartesian
,' coordinates, 100
in wave mechanics,
GUfedimensional clap
Para hydro' sn, 357, 408
Parhelium, $M
Particl^pfl^irjfeOo^^
i in field i vee sp^^jlHL.
t&vniri quantum thlpry, 33
a, F., 222
Kfr»**fcSb series, 43 >
L 58, 112, 209, 210, 219,
f327, 403
iion principle for protons, 357
Mlling, L., 227, 256, 257, 290, 327,
347, 359, 362, 364, 365, 376, 379,
385, 436
Bakeris, C. L., 272
P<4zer, H., 414
Pe&etration, of the core, 213
r nonclassical region, 75
Pe^, W. G., 379
Pe^&Syiaui operator, 231
Perikutatffl» even and odd, defini
tion (
Perturbatfd&Aiause of transitions,
294jf.
definition of. 156
theory of. 15
Perturbation, theory of, firstorder
for a degenerate level, 165
for nondegenerate levels, 156
generalized, 191
involving the time, 294^.
secondorder, 176
approximate, 204
Phase integrals in quantum mechan
ics, 200
Phases of motion, 286
Photochemistry, 26
Photoelectric effect, 25
Photon, 25
Physical constants, values of, 439
Physical interpretation, of harmonic
oscillator functions, 73
of wave equation, 298
of wave functions, 63, 88
Pike, H. H. M., 290
Placzek, G., 290, 293
Planck, M., 25
Planck's constant, 25
Planck's radiation law, 301
Plane rotator. Stark effect of, 177
Podolanski, J., 335
Podolsky, B., 276, 436
Polanyi, M., 374, 388, 414
Polar coordinates, spherical, 9
Polarizability, and dielectric con
stant, 227
of helium atom, 226
of hydrogen atom, 185, 198, 205
and index of refraction, 227
of plane rotator, 178
and van der Waals forces, 387
Polarization, of emitted light, 308
of a gas, 227
Polarization energy, of hydrogen
molecule, 349
of hydrogen moleculeion, 332
Polynomial method of solving wave
equation, 68
Positive states, 392
Postulates of wave mechanics, addi
tional, 298
PoBtulatory basis of physics, 52
Potential energy, average, for hydro
gen atom, 146
464
INDEX
Potential energy, definition of, 2
Potential function for diatomic
molecules, 267
Powerseries method of solving wave
equation, 69
Present, R. D., 353
Principal axes of inertia, definition
of, 275
Probability, of distribution func
tions, 63
of stationary states, 298
of transition, 299
Proper functions, definition of, 58
Properties of wave functions, 58
Quadratic form, minimization of, 203
Quadrupole interaction, 384
Quadrupole moment, definition of,
23
Quantization, rules of, 28
spatial, 45
Quantum of energy, 25
Quantum number, azimuthal, 40,
120
electronspin, 208
A in molecules, 390
magnetic, 40, 117
orbital angular momentum, 237
radial, 124
rotational, 33
spin for atoms, 237
total, 41, 124
Quantum numbers, 87, 124
halfintegral, 48
in wave mechanics, 62
Quantum statistical mechanics, 219,
395jf.
Quantum theory, history of, 25
old, 25jf.
Quantum weight, definition of, 100
R
Rabi, I. I., 276
Rademacher, H.. 276
Radial distribution function for
hydrogen atom, 140
Radial quantum number, 124
Radiation, emission and absorption
of, 21, 299
of kinetic energy, 314
Planck's law of, 301
Rate of chemical reactions, 412
Reaction rates, 412
Recursion formula, definition <ff, 70
for Hermite polynomials, 71
for Legendre polynomials, 126
Reduced mass, 18, 37
Regular point, definition of, 109
Reiche, F., 276
Relativist change of mass of
elep^ T 209
Relati stic doublets, 209
Representation, 423
Repulsion of helium atoms, 361
Repulsive states of hydrogen mol
ecule, 354
Resonance, among bond structures,
377
class\ d, 31*)
dei ^;. a , neo— 
energy, <>*. „ r
frequency oi, 320 v ;
in the hydrog ^n moleculew^ 830
332 ^
integrals, definition of, !
phenomenon, 214 
quantummechanical, 314, 318
Restrictions on wave functions, 58
Richardson, O. W., 336
Richardson, R. G. D., 202
Rigid rotator, wave functions for,
271
Ritz combination principle, 27
Ritz method of solution, 189
Ritz variation method, 189
Robertson, H. P., 105, 432
Robinson, G., 201
Rosen, N., 332, 349 ^
Rotation, of diatomjjf molecules, 2153
of molecules in crystals, 290
of polyatomic ijaolecules, 275
Rotational energy of molecules, 259